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Due to recent innovations to the laser spray technique, a new method of laser ablation using the spray method has surfaced. Laserspray inlet ionization (LSII) involves a matrix/analyte sample at atmospheric pressure being ablated, and the ionization process will take place in an ion transfer capillary tube located in the mass spectrometer inlet. The LSII method is also known as laserspray ionization vacuum (LSIV). | 0 | Chromatography + Titration + pH indicators |
Aaron Klug suggested in 1979 that a technique that was originally developed for structure determination of membrane protein structures can also be used for structure determination of inorganic crystals. This idea was picked up by the research group of Sven Hovmöller which proved that the metal framework partial structure of the KNbWO heavy-metal oxide could be determined from single HREM images recorded at Scherzer defocus. (Scherzer defocus ensures within the weak-phase object approximation a maximal contribution to the image of elastically scattered electrons that were scattered just once while contributions of doubly elastically scattered electrons to the image are optimally suppressed.)
In later years the methods became more sophisticated so that also non-Scherzer images could be processed. One of the most impressive applications at that time was the determination of the complete structure of the complex compound TiSe, which has been inaccessible by X-ray crystallography. Since CIP on single HREM images works only smoothly for layer-structures with at least one short (3 to 5 Å) crystal axis, the method was extended to work also with data from different crystal orientations (= atomic resolution electron tomography). This approach was used in 1990 to reconstruct the 3D structure of the mineral staurolite HFeAlSiO and more recently to determine the structures of the huge quasicrystal approximant phase ν-AlCrFe and the structures of the complex zeolites TNU-9 and IM-5. As mentioned below in the section on crystallographic processing of images that were recorded from 2D periodic arrays with other types of microscopes, the CIP techniques were taken up since 2009 by members of the scanning transmission electron microscopy, scanning probe microscopy and applied crystallography communities.
Contemporary robotics and computer vision researchers also deal with the topic of "computational symmetry", but have so far failed to utilize the spatial distribution of site symmetries that result from crystallographic origin conventions. In addition, a well known statistician noted in his comments on "Symmetry as a continuous feature" that symmetry groups possess inclusion relations (are not disjoint in other words) so that conclusions about which symmetry is most likely present in an image need to be based on "geometric inferences". Such inferences are deeply rooted in information theory, where one is not trying to model empirical data, but extracts and models the information content of the data.
The key difference between geometric inference and all kinds of traditional statistical inferences is that the former merely states the co-existence of a set of definitive (and exact geometrical) constraints and noise, whereby noise is nothing else but an unknown characteristic of the measurement device and data processing operations. From this follows that "in comparing two" (or more) "geometric models we must take into account the fact that the noise is identical (but unknown) and has the same characteristic for both" (all) "models". Because many of these approaches use linear approximations, the level of random noise needs to be low to moderate, or in other words, the measuring devices must be very well corrected for all kinds of known systematic errors.
These kinds of ideas have, however, only been taken up by a tiny minority of researchers within the computational symmetry and scanning probe microscopy / applied crystallography communities.
It is fair to say that the members of computational symmetry community are doing crystallographic image processing under a different name and without utilization of its full mathematical framework (e.g. ignorance to the proper choice of the origin of a unit cell and preference for direct space analyses). Frequently, they are working with artificially created 2D periodic patterns, e.g. wallpapers, textiles, or building decoration in the Moorish/Arabic/Islamic tradition. The goals of these researchers are often related to the identification of point and translation symmetries by computational means and the subsequent classifications of patterns into groups. Since their patterns were artificially created, they do not need to obey all of the restrictions that nature typically imposes on long range periodic ordered arrays of atoms or molecules.
Computational geometry takes a broader view on this issue and concluded already in 1991 that the problem of testing approximate point symmetries in noisy images is in general NP-hard and later on that it is also NP-complete. For restricted versions of this problem, there exist polynomial time algorithms that solve the corresponding optimization problems for a few point symmetries in 2D. | 1 | Crystallography |
The thermometric titrimetric analysis of sodium aluminate liquor ("Bayer liquor") in the production of alumina from bauxite is accomplished in an automated two titration sequence. This is an adaptation of a classic thermometric titration application (VanDalen and Ward, 1973). In the first titration, tartrate solution is added to an aliquot of liquor to complex aluminate, releasing one mole of hydroxyl for each mole of aluminate present. This is titrated acidimetrically along with "free" hydroxyl present and the carbonate content (as a second endpoint). The second titration is preceded by the automatic addition of fluoride solution. The alumina-tartrate complex is broken in favour of the formation of an aluminium fluoride complex and the concomitant release of three moles of hydroxyl for each mole of aluminium present, which are then titrated acidimetrically. The whole determination can be completed in less than 5 minutes. | 0 | Chromatography + Titration + pH indicators |
Vapor diffusion is the most commonly employed method of protein crystallization. In this method, droplets containing purified protein, buffer, and precipitant are allowed to equilibrate with a larger reservoir containing similar buffers and precipitants in higher concentrations. Initially, the droplet of protein solution contains comparatively low precipitant and protein concentrations, but as the drop and reservoir equilibrate, the precipitant and protein concentrations increase in the drop. If the appropriate crystallization solutions are used for a given protein, crystal growth occurs in the drop. This method is used because it allows for gentle and gradual changes in concentration of protein and precipitant concentration, which aid in the growth of large and well-ordered crystals.
Vapor diffusion can be performed in either hanging-drop or sitting-drop format. Hanging-drop apparatus involve a drop of protein solution placed on an inverted cover slip, which is then suspended above the reservoir. Sitting-drop crystallization apparatus place the drop on a pedestal that is separated from the reservoir. Both of these methods require sealing of the environment so that equilibration between the drop and reservoir can occur. | 1 | Crystallography |
In analytical chromatography, the goal is to separate and uniquely identify each of the compounds in a substance. Alternatively, preparative scale chromatography is a method of purification of large batches of material in a production environment. The basic methods of separation in HPLC rely on a mobile phase (water, organic solvents, etc.) being passed through a stationary phase (particulate silica packings, monoliths, etc.) in a closed environment (column); the differences in reactivity among the solvent of interest and the mobile and stationary phases distinguish compounds from one another in a series of adsorption and desorption phenomena. The results are then visually displayed in a resulting chromatogram. Stationary phases are available in many varieties of packing styles as well as chemical structures and can be functionalized for added specificity. Monolithic-style columns, or monoliths, are one of many types of stationary phase structure.
Monoliths, in chromatographic terms, are porous rod structures characterized by mesopores and macropores. These pores provide monoliths with high permeability, a large number of channels, and a high surface area available for reactivity. The backbone of a monolithic column is composed of either an organic or inorganic substrate in, and can easily be chemically altered for specific applications. Their unique structure gives them several physico-mechanical properties that enable them to perform competitively against traditionally packed columns.
Historically, the typical HPLC column consists of high-purity particulate silica compressed into stainless steel tubing. To decrease run times and increase selectivity, smaller diffusion distances have been pursued. To achieve smaller diffusion distances there has been a decrease in the particle sizes. However, as the particle size decreases, the backpressure (for a given column diameter and a given volumetric flow) increases proportionally. Pressure is inversely proportional to the square of the particle size; i.e., when particle size is halved, pressure increases by a factor of four. This is because as the particle sizes get smaller, the interstitial voids (the spaces between the particles) do as well, and it is harder to push the compounds through the smaller spaces. Modern HPLC systems are generally designed to withstand about of backpressure in order to deal with this problem.
Monoliths also have very short diffusion distances, while also providing multiple pathways for solute dispersion. Packed particle columns have pore connectivity values of about 1.5, while monoliths have values ranging from 6 to greater than 10. This means that, in a particulate column, a given analyte may diffuse into and out of the same pore, or enter through one pore and exit through a connected pore. By contrast, an analyte in a monolith is able to enter one channel and exit through any of 6 or more different venues. Little of the surface area in a monolith is inaccessible to compounds in the mobile phase. The high degree of interconnectivity in monoliths confers an advantage seen in the low backpressures and readily achievable high flow rates.
Monoliths are ideally suited for large molecules; although the purification of larger molecules can be very time-consuming. As mentioned previously, particle sizes are decreasing in an attempt to achieve higher resolution and faster separations, which led to higher backpressures. When the smaller particle sizes are used to separate biomolecules, backpressures increase further because of the large molecule size. In monoliths, where backpressures are low and channel sizes are large, small molecule separations are less efficient. This is demonstrated by the dynamic binding capacities, a measure of how much sample can bind to the surface of the stationary phase. Dynamic binding capacities of monoliths for large molecules can be an order of ten times greater than that for particulate packings.
Monoliths exhibit no shear forces or eddying effects. High interconnectivity of the mesopores allows for multiple avenues of convective flow through the column. Mass transport of solutes through the column is relatively unaffected by flow rate. This is completely at odds to traditional particulate packings, whereby eddy effects and shear forces contribute greatly to the loss of resolution and capacity, as seen in the vanDeemter curve. Monoliths can, however, suffer from a different flow disadvantage: wall effects. Silica monoliths, especially, have a tendency to pull away from the sides of their column encasing. When this happens, the flow of the mobile phase occurs around the stationary phase as well as through it, decreasing resolution. Wall effects have been reduced greatly by advances in column construction.
Other advantages of monoliths conferred by their individual construction include greater column to column and batch to batch reproducibility. One technique of creating monolith columns is to polymerize the structure in situ. This involves filling the mold or column tubing with a mixture of monomers, a cross-linking agent, a free-radical initiator, and a porogenic solvent, then initiating the polymerization process under carefully controlled thermal or irradiating conditions. Monolithic in situ polymerization avoids the primary source of column to column variability, which is the packing procedure.
Additionally, packed particle columns must be maintained in a solvent environment and cannot be exposed to air during or after the packing procedure. If exposed to air, the pores dry out and no longer provide adequate surface area for reactivity; the column must be repacked or discarded. Further, because particle compression and packing uniformity are not relevant to monoliths, they exhibit greater mechanical robustness; if particulate columns are dropped, for example, the integrity of the column may be corrupted. Monolithic columns are more physically stable than their particulate counterparts. | 0 | Chromatography + Titration + pH indicators |
There are two equivalent ways to define the meaning of the Miller indices: via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors. Both definitions are given below. In either case, one needs to choose the three lattice vectors a, a, and a that define the unit cell (note that the conventional unit cell may be larger than the primitive cell of the Bravais lattice, as the examples below illustrate). Given these, the three primitive reciprocal lattice vectors are also determined (denoted b, b, and b).
Then, given the three Miller indices denotes planes orthogonal to the reciprocal lattice vector:
That is, (hkℓ) simply indicates a normal to the planes in the basis of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction.
Equivalently, (hkℓ) denotes a plane that intercepts the three points a/h, a/k, and a/ℓ, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane, in the basis of the lattice vectors. If one of the indices is zero, it means that the planes do not intersect that axis (the intercept is "at infinity").
Considering only (hkℓ) planes intersecting one or more lattice points (the lattice planes), the perpendicular distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula: .
The related notation [hkℓ] denotes the direction:
That is, it uses the direct lattice basis instead of the reciprocal lattice. Note that [hkℓ] is not generally normal to the (hkℓ) planes, except in a cubic lattice as described below. | 1 | Crystallography |
A point C on the screw axis satisfies the equation:
Solve this equation for C using Cayley's formula for a rotation matrix
where [B] is the skew-symmetric matrix constructed from Rodrigues' vector
such that
Use this form of the rotation A to obtain
which becomes
This equation can be solved for C on the screw axis P(t) to obtain,
The screw axis of this spatial displacement has the Plücker coordinates . | 1 | Crystallography |
Twinning is crystallographically defined by its twin plane 𝑲, the mirror plane in the twin and parent material, and 𝜼 which is the twinning shear direction. Deformation twins in Zr are generally lenticular in shape, lengthening in the 𝜼 direction and thickening along the 𝑲 plane normal.
The twin plane, shear direction, and shear plane form the basis vectors of an orthogonal set. The axis-angle misorientation relationship between the parent and twin is a rotation of angle 𝜉 about the shear plane's normal direction 𝑷.
More generally, twinning can be described as a 180° rotation about an axis (𝑲 for type I twins or 𝜼 for type II twins normal direction) , or a mirror reflection in a plane (𝑲 or 𝜼 normal plane).
In addition to a homogeneous shear, atomic shuffles are sometimes required to reform the correct crystal structure in the twinned lattice. For each twin variant, a reciprocal twin with swapped 𝑲 and 𝑲 𝜼 and 𝜼 is possible, but one variant may appear more frequently in reality due to complexities with the required shuffles.
there are only two crystallographic planes in a shearing action that do not change their shape and size as a consequence of the shear. The first 𝑲 is the plane defining the upper and lower surfaces of the sheared volume. This plane contains the shear direction. The other plane, designated C. The shear direction is shown with an arrow and labelled with its customary designation 𝜼. It follows from the above that there are three ways that a crystal lattice can be sheared while still retaining its crystal structure and symmetry:
# When 𝑲 is a rational plane and 𝜼 a rational direction, a twin of the first kind
# When 𝑲 is a rational plane and 𝜼 a rational direction, a twin of the second kind, rare
# When all four elements 𝑲, 𝑲, 𝜼, and 𝜼 are rational, a compound twin | 1 | Crystallography |
A layered model of homogeneous and isotropic material, can be up-scaled to a transverse isotropic medium, proposed by Backus.
Backus presented an equivalent medium theory, a heterogeneous medium can be replaced by a homogeneous one that predicts wave propagation in the actual medium. Backus showed that layering on a scale much finer than the wavelength has an impact and that a number of isotropic layers can be replaced by a homogeneous transversely isotropic medium that behaves exactly in the same manner as the actual medium under static load in the infinite wavelength limit.
If each layer is described by 5 transversely isotropic parameters , specifying the matrix
The elastic moduli for the effective medium will be
where
denotes the volume weighted average over all layers.
This includes isotropic layers, as the layer is isotropic if , and . | 1 | Crystallography |
The retardation factor, R, is commonly used in paper chromatography and thin layer chromatography for analyzing and comparing different substances. It can be mathematically described by the following ratio:
An R value will always be in the range 0 to 1; if the substance moves, it can only move in the direction of the solvent flow, and cannot move faster than the solvent. For example, if particular substance in an unknown mixture travels 2.5 cm and the solvent front travels 5.0 cm, the retardation factor would be 0.50. One can choose a mobile phase with different characteristics (particularly polarity) in order to control how far the substance being investigated migrates.
An R value is characteristic for any given compound (provided that the same stationary and mobile phases are used). It can provide corroborative evidence as to the identity of a compound. If the identity of a compound is suspected but not yet proven, an authentic sample of the compound, or standard, is spotted and run on a TLC plate side by side (or on top of each other) with the compound in question. Note that this identity check must be performed on a single plate, because it is difficult to duplicate all the factors which influence R exactly from experiment to experiment. | 0 | Chromatography + Titration + pH indicators |
After the entire sample is loaded, the feed is switched to the displacer, chosen to have higher affinity than any sample component. The displacer forms a sharp-edged zone at the head of the column, pushing the other components downstream. Each sample component now acts as a displacer for the lower-affinity solutes, and the solutes sort themselves out into a series of contiguous bands (a "displacement train"), all moving downstream at the rate set by the displacer. The size and loading of the column are chosen to let this sorting process reach completion before the components reach the bottom of the column. The solutes appear at the bottom of the column as a series of contiguous zones, each consisting of one purified component, with the concentration within each individual zone effectively uniform. | 0 | Chromatography + Titration + pH indicators |
Asbestiform is a crystal habit. It describes a mineral that grows in a fibrous aggregate of high tensile strength, flexible, long, and thin crystals that readily separate. The most common asbestiform mineral is chrysotile, commonly called "white asbestos", a magnesium phyllosilicate part of the serpentine group. Other asbestiform minerals include riebeckite, an amphibole whose fibrous form is known as crocidolite or "blue asbestos", and brown asbestos, a cummingtonite-grunerite solid solution series.
The United States Environmental Protection Agency explains that, "In general, exposure may occur only when the asbestos-containing material is disturbed or damaged in some way to release particles and fibers into the air."
"Mountain leather" is an old-fashioned term for flexible, sheet-like natural formations of asbestiform minerals which resemble leather. Asbestos-containing minerals known to form mountain leather include: actinolite, palygorskite, saponite, sepiolite, tremolite, and zeolite. | 1 | Crystallography |
By combining topographic image formation with tomographic image reconstruction, distributions of defects can be resolved in three dimensions. Unlike "classical" computed tomography (CT), image contrast is not based on differences in absorption (absorption contrast), but on the usual contrast mechanisms of topography (diffraction contrast). In this way, three-dimensional distributions of dislocations in crystals have been imaged.
Literature: | 1 | Crystallography |
Phase transformation crystallography describes the orientation relationship and interface orientation after a phase transformation (such as martensitic transformation or precipitation). | 1 | Crystallography |
Congo red was first synthesized in 1883 by Paul Böttiger, who had been employed at Friedrich Bayer Company in Elberfeld, Germany. He was looking for textile dyes that did not require a mordant step. The company which had a right of first refusal to his inventions was not interested in this bright red color, so he filed the patent under his own name and sold it to the AGFA company of Berlin. AGFA marketed the dye under the name "Congo red", a catchy name in Germany at the time of the 1884 Berlin West Africa Conference, an important event in the Colonisation of Africa. The dye was a major commercial success for AGFA. In the following years, for the same reason, other dyes were marketed using the "Congo" name: Congo rubine, Congo corinth, brilliant Congo, Congo orange, Congo brown, and Congo blue. Once of economic significance, Congo red has fallen into disuse as have all benzidine-derived dyes, owing to their carcinogenic activity.
It is prepared by azo coupling of the bis(diazonium) derivative of benzidine with naphthionic acid.
Congo blue, however, is in widespread international use, in gel sheet form, as a filter to place in front of theatrical, motion picture, television, church, and live event lighting instruments. It is sold under the item name "181 Congo Blue" by Lee Filters. It emits a deep rich saturated blue color with elements of red. Depending upon the color temperature of the source lamp, the light from a lighting instrument with a Congo Blue filter reflected from a white surface can vary from very saturated blue to purple or violet. The manufacturer reports that fluorescent light through a Congo Blue filter gives the appearance of black light. Congo Blue filters are frequently used at live music concerts at an angle from behind musicians to cross back-light with a "warm" color gel like yellow, straw, gold, orange, or magenta, from an opposing angle, for a very dramatic effect. Another use of Congo Blue filters by lighting technicians, is to cut a small strip from the gel sheet, which the technician looks through to make brightness adjustments to a video monitor displaying a standard color bar chart. The Congo Blue filter effectively removes the color from chart and shows the separate bars only in terms of their differing incremental brightness levels. This allows the technician to adjust the monitor to show a full and correct range of brightnesses. | 0 | Chromatography + Titration + pH indicators |
Early records of the discovery of polymorphism credit Eilhard Mitscerlich and Jöns Jacob Berzelius for their studies of phosphates and arsenates in the early 1800s. The studies involved measuring the interfacial angles of the crystals to show that chemically identical salts could have two different forms. Mitscerlich originally called this discovery isomorphism. The measurement of crystal density was also used by Wilhelm Ostwald and expressed in Ostwald's Ratio.
The development of the microscope enhanced observations of polymorphism and aided Moritz Ludwig Frankenheim’s studies in the 1830s. He was able to demonstrate methods to induce crystal phase changes and formally summarized his findings on the nature of polymorphism. Soon after, the more sophisticated polarized light microscope came into use, and it provided better visualization of crystalline phases allowing crystallographers to distinguish between different polymorphs. The hot stage was invented and fitted to a polarized light microscope by Otto Lehmann in about 1877. This invention helped crystallographers determine melting points and observe polymorphic transitions.
While the use of hot stage microscopes continued throughout the 1900s, thermal methods also became commonly used to observe the heat flow that occurs during phase changes such as melting and polymorphic transitions. One such technique, differential scanning calorimetry (DSC), continues to be used for determining the enthalpy of polymorphic transitions.
In the 20th century, X-ray crystallography became commonly used for studying the crystal structure of polymorphs. Both single crystal x-ray diffraction and powder x-ray diffraction techniques are used to obtain measurements of the crystal unit cell. Each polymorph of a compound has a unique crystal structure. As a result, different polymorphs will produce different x-ray diffraction patterns.
Vibrational spectroscopic methods came into use for investigating polymorphism in the second half of the twentieth century and have become more commonly used as optical, computer, and semiconductor technologies improved. These techniques include infrared (IR) spectroscopy, terahertz spectroscopy and Raman spectroscopy. Mid-frequency IR and Raman spectroscopies are sensitive to changes in hydrogen bonding patterns. Such changes can subsequently be related to structural differences. Additionally, terahertz and low frequency Raman spectroscopies reveal vibrational modes resulting from intermolecular interactions in crystalline solids. Again, these vibrational modes are related to crystal structure and can be used to uncover differences in 3-dimensional structure among polymorphs. | 1 | Crystallography |
According to Ostwald's rule, usually less stable polymorphs crystallize before the stable form. The concept hinges on the idea that unstable polymorphs more closely resemble the state in solution, and thus are kinetically advantaged. The founding case of fibrous vs rhombic benzamide illustrates the case. Another example is provided by two polymorphs of titanium dioxide. Nevertheless, there are known systems, such as metacetamol, where only narrow cooling rate favors obtaining metastable form II.
Polymorphs have disparate stabilities. Some convert rapidly at room (or any) temperature. Most polymorphs of organic molecules only differ by a few kJ/mol in lattice energy. Approximately 50% of known polymorph pairs differ by less than 2 kJ/mol and stability differences of more than 10 kJ/mol are rare. Valuable to mention that polymorph stability may change upon temperature or pressure. Important to note that structural and thermodybnamic stability are different. Thermodynamic stability may be studied using experimental or computational methods.
Polymorphism is affected by the details of crystallisation. The solvent in all respects affects the nature of the polymorph, including concentration, other components of the solvent, i.e., species that inhibiting or promote certain growth patterns. A decisive factor is often the temperature of the solvent from which crystallisation is carried out.
Metastable polymorphs are not always reproducibly obtained, leading to cases of "disappearing polymorphs", with usually negative implications on law and business. | 1 | Crystallography |
In this technique a square or rectangular paper is used. Here the sample is applied to one of the corners and development is performed at a right angle to the direction of the first run. | 0 | Chromatography + Titration + pH indicators |
Discrete misorientations or the misorientation distribution can be fully described as plots in the Euler angle, axis/angle, or Rodrigues vector space. Unit quaternions, while computationally convenient, do not lend themselves to graphical representation because of their four-dimensional nature. For any of the representations, plots are usually constructed as sections through the fundamental zone; along φ in Euler angles, at increments of rotation angle for axis/angle, and at constant ρ (parallel to <001>) for Rodrigues. Due to the irregular shape of the cubic-cubic FZ, the plots are typically given as sections through the cubic FZ with the more restrictive boundaries overlaid.<br />
<br />
Mackenzie plots are a one-dimensional representation of the MD plotting the relative frequency of the misorientation angle, irrespective of the axis. Mackenzie determined the misorientation distribution for a cubic sample with a random texture. | 1 | Crystallography |
In physics, the reciprocal lattice emerges from the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system (usually a Bravais lattice). The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, where refers to the wavevector.
In quantum physics, reciprocal space is closely related to momentum space according to the proportionality , where is the momentum vector and is the reduced Planck constant. The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively.
The reciprocal lattice is the set of all vectors , that are wavevectors of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice . Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of at each direct lattice point (so essentially same phase at all the direct lattice points).
The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In neutron, helium and X-ray diffraction, due to the Laue conditions, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal.
The Brillouin zone is a Wigner–Seitz cell of the reciprocal lattice. | 1 | Crystallography |
The effect of crystal symmetry on misorientations is to reduce the fraction of the full orientation space necessary to uniquely represent all possible misorientation relationships. For example, cubic crystals (i.e. FCC) have 24 symmetrically related orientations. Each of these orientations is physically indistinguishable, though mathematically distinct. Therefore, the size of orientation space is reduced by a factor of 24. This defines the fundamental zone (FZ) for cubic symmetries. For the misorientation between two cubic crystallites, each possesses its 24 inherent symmetries. In addition, there exists a switching symmetry, defined by:
which recognizes the invariance of misorientation to direction; A→B or B→A. The fraction of the total orientation space in the cubic-cubic fundamental zone for misorientation is then given by:<br />
or 1/48 the volume of the cubic fundamental zone. This also has the effect of limiting the maximum unique misorientation angle to 62.8°<br />
<br />
Disorientation describes the misorientation with the smallest possible rotation angle out of all symmetrically equivalent misorientations that fall within the FZ (usually specified as having an axis in the standard stereographic triangle for cubics). Calculation of these variants involves application of crystal symmetry operators to each of the orientations during the calculation of misorientation.<br />
<br />
where O denotes one of the symmetry operators for the material. | 1 | Crystallography |
The width of the diffraction peaks are found to broaden at higher Bragg angles. This angular dependency was originally represented by
where , , and are the half-width parameters and may be refined during the fit. | 1 | Crystallography |
To make a geometrically stable structure in a mineral, atoms must fit together in terms of both their size and charge. The atoms have to fit together so that their electron shells can interact with one another and they also have to produce a neutral molecule. For these reasons the sizes and electron shell structure of atoms determine what element combinations are possible and the geometrical form that various minerals take. Because electrons are donated and received, it is the ionic radius of the element that controls the size and determines how atoms fit together in minerals. | 1 | Crystallography |
In Cartesian coordinates the 2 basis vectors are represented by a cell tensor :
The area of the unit cell, , is given by the determinant of the cell matrix:
For the special case of a square or rectangular unit cell, the matrix is diagonal, and we have that: | 1 | Crystallography |
Cyanidin is a natural organic compound. It is a particular type of anthocyanidin (glycoside version called anthocyanins). It is a pigment found in many red berries including grapes, bilberry, blackberry, blueberry, cherry, chokeberry, cranberry, elderberry, hawthorn, loganberry, açai berry and raspberry. It can also be found in other fruits such as apples and plums, and in red cabbage and red onion. It has a characteristic reddish-purple color, though this can change with pH; solutions of the compound are red at pH < 3, violet at pH 7-8, and blue at pH > 11. In certain fruits, the highest concentrations of cyanidin are found in the seeds and skin. Cyanidin has been found to be a potent sirtuin 6 (SIRT6) activator. | 0 | Chromatography + Titration + pH indicators |
An example of a transversely isotropic material is the so-called on-axis unidirectional fiber composite lamina where the fibers are circular in cross section. In a unidirectional composite, the plane normal to the fiber direction can be considered as the isotropic plane, at long wavelengths (low frequencies) of excitation. In the figure to the right, the fibers would be aligned with the axis, which is normal to the plane of isotropy.
In terms of effective properties, geological layers of rocks are often interpreted as being transversely isotropic. Calculating the effective elastic properties of such layers in petrology has been coined Backus upscaling, which is described below. | 1 | Crystallography |
The development of stable red color in the surface of the medium indicates sufficient acid production to lower the pH to 4.4 and constitute a positive test. Since other organism may produce lesser quantities of acid from the test substrate, an intermediate orange color between yellow and red may develop. This does not indicate positive test. | 0 | Chromatography + Titration + pH indicators |
The moving-belt interface (MBI) was developed by McFadden et al. in 1977 and commercialized by Finnigan. This interface consisted of an endless moving belt onto which the LC column effluent was deposited in a band. On the belt, the solvent was evaporated by gently heating and efficiently exhausting the solvent vapours under reduced pressure in two vacuum chambers. After the liquid phase was removed, the belt passed over a heater which flash desorbed the analytes into the MS ion source. One of the significant advantages of the MBI was its compatibility with a wide range of chromatographic conditions. MBI was successfully used for LC–MS applications between 1978 and 1990 because it allowed coupling of LC to MS devices using EI, CI, and fast-atom bombardment (FAB) ion sources. The most common MS systems connected by MBI interfaces to LC columns were magnetic sector and quadrupole instruments. MBI interfaces for LC–MS allowed MS to be widely applied in the analysis of drugs, pesticides, steroids, alkaloids, and polycyclic aromatic hydrocarbons. This interface is no longer used because of its mechanical complexity and the difficulties associated with belt renewal (or cleaning) as well as its inability to handle very labile biomolecules. | 0 | Chromatography + Titration + pH indicators |
The topographic image of a uniform crystal with a perfectly regular lattice, illuminated by a homogeneous beam, is uniform (no contrast). Contrast arises when distortions of the lattice (defects, tilted crystallites, strain) occur; when the crystal is composed of several different materials or phases; or when the thickness of the crystal changes across the image domain. | 1 | Crystallography |
Thermal ellipsoids, more formally termed atomic displacement parameters or anisotropic displacement parameters, are ellipsoids used in crystallography to indicate the magnitudes and directions of the thermal vibration of atoms in crystal structures. Since the vibrations are usually anisotropic (different magnitudes in different directions in space), an ellipsoid is a convenient way of visualising the vibration and therefore the symmetry and time averaged position of an atom in a crystal. Their theoretical framework was introduced by D. W. J. Cruickshank in 1956 and the concept was popularized through the program ORTEP (Oak Ridge Thermal-Ellipsoid Plot Program), first released in 1965.
Thermal ellipsoids can be defined by a tensor, a mathematical object which allows the definition of magnitude and orientation of vibration with respect to three mutually perpendicular axes. The three principal axes of the thermal vibration of an atom are denoted , , and , and the corresponding thermal ellipsoid is based on these axes. The size of the ellipsoid is scaled so that it occupies the space in which there is a particular probability of finding the electron density of the atom. The particular probability is usually 50%. | 1 | Crystallography |
The plate height given as:
with the column length and the number of theoretical plates can be estimated from a chromatogram by analysis of the retention time for each component and its standard deviation as a measure for peak width, provided that the elution curve represents a Gaussian curve.
In this case the plate count is given by:
By using the more practical peak width at half height the equation is:
or with the width at the base of the peak: | 0 | Chromatography + Titration + pH indicators |
Energy-dispersive x-ray spectroscopy (EDS) and electron energy loss spectroscopy (EELS) are commonly used techniques to both qualitatively and quantitatively probe the composition of samples in the TEM. A primary challenge in the quantitative accuracy of both techniques is the phenomenon of channelling. Put simply, in a crystalline solid, the probability of interaction between an electron and ion in the lattice depends strongly on the momentum (direction and velocity) of the electron. When probing a sample under diffraction conditions near a zone axis, as is often the case in EDS and EELS applications, channelling can have a large impact on the effective interaction of the incident electrons with specific ions in the crystal structure. In practice, this can lead to erroneous measurements of composition that depend strongly on the orientation and thickness of the sample and the accelerating voltage. Since PED entails an integration over incident directions of the electron probe, and generally does not include beams parallel to the zone axis, the detrimental channeling effects outlined above can be minimized, yielding far more accurate composition measurements in both techniques. | 1 | Crystallography |
The Pearson symbol, or Pearson notation, is used in crystallography as a means of describing a crystal structure, and was originated by W. B. Pearson. The symbol is made up of two letters followed by a number. For example:
* Diamond structure, cF8
* Rutile structure, tP6
The two (italicised) letters specify the Bravais lattice. The lower-case letter specifies the crystal family, and the upper-case letter the centering type. The number at the end of the Pearson symbol gives the number of the atoms in the conventional unit cell.
The letters A, B and C were formerly used instead of S. When the centred face cuts the X axis, the Bravais lattice is called A-centred. In analogy, when the centred face cuts the Y or Z axis, we have B- or C-centring respectively.
The fourteen possible Bravais lattices are identified by the first two letters: | 1 | Crystallography |
Acid orange 5 may be prepared by diazotization of sulfanilic acid, followed by reaction with diphenylamine: | 0 | Chromatography + Titration + pH indicators |
Time crystals seem to break time-translation symmetry and have repeated patterns in time even if the laws of the system are invariant by translation of time. The time crystals that are experimentally realized show discrete time-translation symmetry breaking, not the continuous one: they are periodically driven systems oscillating at a fraction of the frequency of the driving force. (According to Philip Ball, DTC are so-called because "their periodicity is a discrete, integer multiple of the driving period".)
The initial symmetry, which is the discrete time-translation symmetry () with , is spontaneously broken to the lower discrete time-translation symmetry with , where is time, the driving period, an integer.
Many systems can show behaviors of spontaneous time-translation symmetry breaking but may not be discrete (or Floquet) time crystals: convection cells, oscillating chemical reactions, aerodynamic flutter, and subharmonic response to a periodic driving force such as the Faraday instability, NMR spin echos, parametric down-conversion, and period-doubled nonlinear dynamical systems.
However, discrete (or Floquet) time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking:
* it is a broken symmetry the system shows oscillations with a period longer than the driving force,
* the system is in crypto-equilibrium these oscillations generate no entropy, and a time-dependent frame can be found in which the system is indistinguishable from an equilibrium when measured stroboscopically (which is not the case of convection cells, oscillating chemical reactions and aerodynamic flutter),
* the system exhibits long-range order the oscillations are in phase (synchronized) over arbitrarily long distances and time.
Moreover, the broken symmetry in time crystals is the result of many-body interactions: the order is the consequence of a collective process, just like in spatial crystals. This is not the case for NMR spin echos.
These characteristics makes discrete time crystals analogous to spatial crystals as described above and may be considered a novel type or phase of nonequilibrium matter. | 1 | Crystallography |
For a face-centered cubic unit cell, the number of atoms is four. A line can be drawn from the top corner of a cube diagonally to the bottom corner on the same side of the cube, which is equal to 4r. Using geometry, and the side length, a can be related to r as:
Knowing this and the formula for the volume of a sphere, it becomes possible to calculate the APF as follows: | 1 | Crystallography |
Orbifold notation for wallpaper groups, advocated by John Horton Conway (Conway, 1992) (Conway 2008), is based not on crystallography, but on topology. One can fold the infinite periodic tiling of the plane into its essence, an orbifold, then describe that with a few symbols.
*A digit, n, indicates a centre of n-fold rotation corresponding to a cone point on the orbifold. By the crystallographic restriction theorem, n must be 2, 3, 4, or 6.
*An asterisk, *, indicates a mirror symmetry corresponding to a boundary of the orbifold. It interacts with the digits as follows:
*#Digits before * denote centres of pure rotation (cyclic).
*#Digits after * denote centres of rotation with mirrors through them, corresponding to "corners" on the boundary of the orbifold (dihedral).
*A cross, ×, occurs when a glide reflection is present and indicates a crosscap on the orbifold. Pure mirrors combine with lattice translation to produce glides, but those are already accounted for so need no notation.
*The "no symmetry" symbol, o, stands alone, and indicates there are only lattice translations with no other symmetry. The orbifold with this symbol is a torus; in general the symbol o denotes a handle on the orbifold.
The group denoted in crystallographic notation by cmm will, in Conway's notation, be 2*22. The 2 before the * says there is a 2-fold rotation centre with no mirror through it. The * itself says there is a mirror. The first 2 after the * says there is a 2-fold rotation centre on a mirror. The final 2 says there is an independent second 2-fold rotation centre on a mirror, one that is not a duplicate of the first one under symmetries.
The group denoted by pgg will be 22×. There are two pure 2-fold rotation centres, and a glide reflection axis. Contrast this with pmg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold.
Coxeter's bracket notation is also included, based on reflectional Coxeter groups, and modified with plus superscripts accounting for rotations, improper rotations and translations. | 1 | Crystallography |
Litmus is a water-soluble mixture of different dyes extracted from lichens. It is often absorbed onto filter paper to produce one of the oldest forms of pH indicator, used to test materials for acidity. In an acidic medium, blue litmus paper turns red, while in a basic or alkaline medium, red litmus paper turns blue. In short, it is a dye and indicator which is used to place substances on a pH scale. | 0 | Chromatography + Titration + pH indicators |
The first potentiometric titration was carried out in 1893 by Robert Behrend at Ostwald's Institute in Leipzig. He titrated mercurous solution with potassium chloride, potassium bromide, and potassium iodide. He used a mercury electrode along with a mercury/mercurous nitrate reference electrode. He found that in a cell composed of mercurous nitrate and mercurous nitrate/mercury, the initial voltage is 0. If potassium chloride is added to mercurous nitrate on one side, mercury (I) chloride is precipitated. This decreased the osmotic pressure of mercury (I) ions on the side and creates a potential difference. This potential difference increases slowly as additional potassium chloride is added, but then increases more rapidly. He found the greatest potential difference is achieved once all of the mercurous nitrate has been precipitated. This was used to discern end points of titrations.
Wilhelm Böttger then developed the tool of potentiometric titration while working at Ostwald's Institute. He used potentiometric titration to observe the differences in titration between strong and weak acids, as well as the behavior of polybasic acids. He introduced the idea of using potentiometric titrations for acids and bases that could not be titrated in conjunction with a colorimetric indicator
Potentiometric titrations were first used for redox titrations by Crotogino. He titrated halide ions with potassium permanganate using a shiny platinum electrode and a calomel electrode. He said that if an oxidizing agent is added to a reducing solution then the equilibrium between the reducing substance and reaction product will shift towards the reaction product. This changes the potential very slowly until the amount of reducing substance becomes very small. A large change in potential will occur then once a small addition of the titrating solution is added, as the final amounts of reducing agent are removed and the potential corresponds solely to the oxidizing agent. This large increase in potential difference signifies the endpoint of the reaction. | 0 | Chromatography + Titration + pH indicators |
A use of ion chromatography can be seen in argentation chromatography. Usually, silver and compounds containing acetylenic and ethylenic bonds have very weak interactions. This phenomenon has been widely tested on olefin compounds. The ion complexes the olefins make with silver ions are weak and made based on the overlapping of pi, sigma, and d orbitals and available electrons therefore cause no real changes in the double bond. This behavior was manipulated to separate lipids, mainly fatty acids from mixtures in to fractions with differing number of double bonds using silver ions. The ion resins were impregnated with silver ions, which were then exposed to various acids (silicic acid) to elute fatty acids of different characteristics.
Detection limits as low as 1 μM can be obtained for alkali metal ions.
It may be used for measurement of HbA1c, porphyrin and with water purification. Ion Exchange Resins(IER) have been widely used especially in medicines due to its high capacity and the uncomplicated system of the separation process. One of the synthetic uses is to use Ion Exchange Resins for kidney dialysis. This method is used to separate the blood elements by using the cellulose membraned artificial kidney.
Another clinical application of ion chromatography is in the rapid anion exchange chromatography technique used to separate creatine kinase (CK) isoenzymes from human serum and tissue sourced in autopsy material (mostly CK rich tissues were used such as cardiac muscle and brain). These isoenzymes include MM, MB, and BB, which all carry out the same function given different amino acid sequences. The functions of these isoenzymes are to convert creatine, using ATP, into phosphocreatine expelling ADP. Mini columns were filled with DEAE-Sephadex A-50 and further eluted with tris- buffer sodium chloride at various concentrations (each concentration was chosen advantageously to manipulate elution). Human tissue extract was inserted in columns for separation. All fractions were analyzed to see total CK activity and it was found that each source of CK isoenzymes had characteristic isoenzymes found within. Firstly, CK- MM was eluted, then CK-MB, followed by CK-BB. Therefore, the isoenzymes found in each sample could be used to identify the source, as they were tissue specific.
Using the information from results, correlation could be made about the diagnosis of patients and the kind of CK isoenzymes found in most abundant activity. From the finding, about 35 out of 71 patients studied suffered from heart attack (myocardial infarction) also contained an abundant amount of the CK-MM and CK-MB isoenzymes. Findings further show that many other diagnosis including renal failure, cerebrovascular disease, and pulmonary disease were only found to have the CK-MM isoenzyme and no other isoenzyme. The results from this study indicate correlations between various diseases and the CK isoenzymes found which confirms previous test results using various techniques. Studies about CK-MB found in heart attack victims have expanded since this study and application of ion chromatography. | 0 | Chromatography + Titration + pH indicators |
The condition on linearly independent translations means that there exist linearly independent vectors v and w (in R) such that the group contains both T and T.
The purpose of this condition is to distinguish wallpaper groups from frieze groups, which possess a translation but not two linearly independent ones, and from two-dimensional discrete point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast to frieze groups, which only repeat along a single axis.
(It is possible to generalise this situation. One could for example study discrete groups of isometries of R with m linearly independent translations, where m is any integer in the range 0 ≤ m ≤ n.) | 1 | Crystallography |
The orientation of a glide plane is given by the position of the symbol in the Hermann–Mauguin designation, just as with mirror planes.
They are noted by a, b, or c depending on which axis (direction) the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure. In cases where there are two possibilities among a, b, and c (such as a or b), the letter e is used. (In these cases, centering entails that both glides occur.) To summarize:
* a, b, or c glide translation along half the lattice vector of this face.
* n glide translation along half a face diagonal.
* d glide planes with translation along a quarter of a face diagonal or of a space diagonal.
* e two glides with the same glide plane and translation along two (different) half-lattice vectors. | 1 | Crystallography |
GC–MS is used for the analysis of unknown organic compound mixtures. One critical use of this technology is the use of GC–MS to determine the composition of bio-oils processed from raw biomass. GC–MS is also utilized in the identification of continuous phase component in a smart material, magnetorheological (MR) fluid. | 0 | Chromatography + Titration + pH indicators |
The formula for dimensions can be derived assuming an -dimensional real vector space with a basis and an inner product . The reciprocal lattice vectors are uniquely determined by the formula . Using the permutation
they can be determined with the following formula:
Here, is the volume form, is the inverse of the vector space isomorphism defined by and denotes the inner multiplication.
One can verify that this formula is equivalent to the known formulas for the two- and three-dimensional case by using the following facts: In three dimensions, and in two dimensions, , where is the rotation by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation). | 1 | Crystallography |
The popularity of the Karl Fischer titration (henceforth referred to as KF) is due in large part to several practical advantages that it holds over other methods of moisture determination, such as accuracy, speed and selectivity.
KF is selective for water, because the titration reaction itself consumes water. In contrast, measurement of mass loss on drying will detect the loss of any volatile substance. However, the strong redox chemistry () means that redox-active sample constituents may react with the reagents. For this reason, KF is unsuitable for solutions containing e.g. dimethyl sulfoxide.
KF has a high accuracy and precision, typically within 1% of available water, e.g. 3.00% appears as 2.97–3.03%. Although KF is a destructive analysis, the sample quantity is small and is typically limited by the accuracy of weighing. For example, in order to obtain an accuracy of 1% using a scale with the typical accuracy of 0.2 mg, the sample must contain 20 mg water, which is e.g. 200 mg for a sample with 10% water. For coulometers, the measuring range is from 1–5 ppm to about 5%. Volumetric KF readily measures samples up to 100%, but requires impractically large amounts of sample for analytes with less than 0.05% water. The KF response is linear. Therefore, single-point calibration using a calibrated 1% water standard is sufficient and no calibration curves are necessary.
Little sample preparation is needed: a liquid sample can usually be directly injected using a syringe. The analysis is typically complete within a minute. However, KF suffers from an error called drift, which is an apparent water input that can confuse the measurement. The glass walls of the vessel adsorb water, and if any water leaks into the cell, the slow release of water into the titration solution can continue for a long time. Therefore, before measurement, it is necessary to carefully dry the vessel and run a 10–30-minute "dry run" in order to calculate the rate of drift. The drift is then subtracted from the result.
KF is suitable for measuring liquids and, with special equipment, gases. The major disadvantage with solids is that the water has to be accessible and easily brought into methanol solution. Many common substances, especially foods such as chocolate, release water slowly and with difficulty, requiring additional efforts to reliably bring the total water content into contact with the Karl Fischer reagents. For example, a high-shear mixer may be installed to the cell in order to break the sample. KF has problems with compounds with strong binding to water, as in water of hydration, for example with lithium chloride, so KF is unsuitable for the special solvent LiCl/DMAc.
KF is suitable for automation. Generally, KF is conducted using a separate KF titrator, or for volumetric titration, a KF titration cell installed into a general-purpose titrator. There are also oven attachments that can be used for materials that have problems being analyzed normally in the cell. The important aspect about the oven attachment is that the material doesn't decompose into water when heated to release the water. The oven attachment also supports automation of samples.
Using volumetric titration with visual detection of a titration endpoint is also possible with coloured samples by UV/VIS spectrophotometric detection. | 0 | Chromatography + Titration + pH indicators |
Common crystals exhibit broken translation symmetry: they have repeated patterns in space and are not invariant under arbitrary translations or rotations. The laws of physics are unchanged by arbitrary translations and rotations. However, if we hold fixed the atoms of a crystal, the dynamics of an electron or other particle in the crystal depend on how it moves relative to the crystal, and particle momentum can change by interacting with the atoms of a crystal — for example in Umklapp processes. Quasimomentum, however, is conserved in a perfect crystal.
Time crystals show a broken symmetry analogous to a discrete space-translation symmetry breaking. For example, the molecules of a liquid freezing on the surface of a crystal can align with the molecules of the crystal, but with a pattern less symmetric than the crystal: it breaks the initial symmetry. This broken symmetry exhibits three important characteristics:
* the system has a lower symmetry than the underlying arrangement of the crystal,
* the system exhibits spatial and temporal long-range order (unlike a local and intermittent order in a liquid near the surface of a crystal),
* it is the result of interactions between the constituents of the system, which align themselves relative to each other. | 1 | Crystallography |
In selective ion monitoring (SIM) certain ion fragments are entered into the instrument method and only those mass fragments are detected by the mass spectrometer. The advantages of SIM are that the detection limit is lower since the instrument is only looking at a small number of fragments (e.g. three fragments) during each scan. More scans can take place each second. Since only a few mass fragments of interest are being monitored, matrix interferences are typically lower. To additionally confirm the likelihood of a potentially positive result, it is relatively important to be sure that the ion ratios of the various mass fragments are comparable to a known reference standard. | 0 | Chromatography + Titration + pH indicators |
CrysTBox is freely available on demand for non-commercial use by non-commercial subjects. The only safe way to download CrysTBox installers is via a request form on the official website. Commercial use is not allowed due to the license of MATLAB used for CrysTBox compilation. | 1 | Crystallography |
Crystal field excitation is the electronic transition of an electron between two orbitals of an atom that is situated in a crystal field environment. They are often observed in coordination complexes of transition metals. Some examples of crystal field excitations are dd-transitions on a copper atom that is surrounded by an octahedron of oxygen atoms, or ff-transitions on the uranium atom in uranium antimonide. | 1 | Crystallography |
The research that appeared to spark an onslaught of modified applications was a gel permeation chromatography technique of fixing poly(isopropyl acrylate) (PIPA) strands to glass beads and separating a mixture of dextrans, which was developed by Gewehr et al. They found that between the temperatures of 25–32 °C, the elution time of dextrans at different molecular weights exhibited a dependence on the temperature. Dextrans of the highest molecular weight eluted first since the PIPA chains exhibit hydrophilicity at temperatures below the LCST. As the temperature of the elution increased, when the chains behave in a more hydrophobic manner, the elution times increased for each of the analytes for the given range. The trend generally applies over the entire temperature range, but there is a flattening of the curve before 25 °C and after 32 °C (the approximate LCST for this experiment). It is important to note that above the LCST, the PIPA acts as a typical nonpolar stationary phase that would be used in reverse-phased chromatography. There are also instances of the elution times increasing below 15 °C, which most likely can be attributed to the lower temperatures’ effects on mass transfer playing a more significant role on retention than the stationary phase behavior. This study showed that the resolution could essentially be tuned by adjusting the operating temperature. The scope of this study was limited to isothermal conditions and attaching polymer chains to glass beads. The results, however, were satisfying enough to inspire other investigations and modifications to create a more versatile stationary phase for the advancement of chromatography. | 0 | Chromatography + Titration + pH indicators |
Euhedral crystals (also known as idiomorphic or automorphic crystals) are those that are well-formed, with sharp, easily recognised faces. The opposite is anhedral (also known as xenomorphic or allotriomorphic): a rock with an anhedral texture is composed of mineral grains that have no well-formed crystal faces or cross-section shape in thin section. Anhedral crystal growth occurs in a competitive environment with no free space for the formation of crystal faces. An intermediate texture with some crystal face-formation is termed subhedral (also known as hypidiomorphic or hypautomorphic).
Crystals that grow from cooling liquid magma typically do not form smooth faces or sharp crystal outlines. As magma cools, the crystals grow and eventually touch each other, preventing crystal faces from forming properly or at all.
When snowflakes crystallize, they do not touch each other. Thus, snowflakes form euhedral, six-sided twinned crystals. In rocks, the presence of euhedral crystals may signify that they formed early in the crystallization of liquid magma or perhaps crystallized in a cavity or vug, without steric hindrance, or spatial restrictions, from other crystals. | 1 | Crystallography |
This machine uses a motor-driven plate to hold a precisely flat disk (known as a "lap") for the purpose of cutting or polishing. Diamond abrasives bonded to metal or resin are typically used for cutting laps, and a wide variety of materials are used for polishing laps in conjunction with either very fine diamond powder or oxide-based polishes. Water is typically used for cutting, while either oil or water is used for the polishing process.
The machine uses a system generally called a "mast" which consists of an angle readout, height adjustment and typically a gear (called an "index gear") with a particular number of teeth is used as a means of setting the rotational angle. The angles of rotation are evenly divided by the number of teeth present on the gear, though many machines include additional means of adjusting the rotational angle in finer increments, often called a "cheater". The stone is bonded to a (typically metal) rod known as a "dop" or "dop stick" and is held in place by part of the mast referred to as the "quill". | 1 | Crystallography |
Partition chromatography theory and practice was introduced through the work and publications of Archer Martin and Richard Laurence Millington Synge during the 1940s. They would later receive the 1952 Nobel Prize in Chemistry "for their invention of partition chromatography". | 0 | Chromatography + Titration + pH indicators |
The term methyl violet encompasses three compounds that differ in the number of methyl groups attached to the amine functional group. They are all soluble in water, ethanol, diethylene glycol and dipropylene glycol. | 0 | Chromatography + Titration + pH indicators |
A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (in other words, shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns that repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.
The types of transformations that are relevant here are called Euclidean plane isometries. For example:
* If one shifts example B one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern is exactly the same as the starting pattern. This type of symmetry is called a translation. Examples A and C are similar, except that the smallest possible shifts are in diagonal directions.
* If one turns example B clockwise by 90°, around the centre of one of the squares, again one obtains exactly the same pattern. This is called a rotation. Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C.
* One can also flip example B across a horizontal axis that runs across the middle of the image. This is called a reflection. Example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A.
However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonal axes. If one flips across a diagonal line, one does not get the same pattern back, but the original pattern shifted across by a certain distance. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C.
Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection. | 1 | Crystallography |
The following are synthetic gemstones that were developed by Tairus scientists; they are alternately referred to as Tairus stones (e.g. "Tairus Ruby").
*Floating zone ruby, synthesized in 1991 (no longer in production)
*Hydrothermal ruby, synthesized in 1992 by Alexander Dokukin.
*Hydrothermal aquamarine, synthesized in 1993.
*Hydrothermal sapphires, in pink, green, orange, and blue.
*Colombian color emerald, developed in 2004. | 1 | Crystallography |
Some authors define stereographic projection from the north pole (0, 0, 1) onto the plane , which is tangent to the unit sphere at the south pole (0, 0, −1). This can be described as a composition of a projection onto the equatorial plane described above, and a homothety from it to the polar plane. The homothety scales the image by a factor of 2 (a ratio of a diameter to a radius of the sphere), hence the values and produced by this projection are exactly twice those produced by the equatorial projection described in the preceding section. For example, this projection sends the equator to the circle of radius 2 centered at the origin. While the equatorial projection produces no infinitesimal area distortion along the equator, this pole-tangent projection instead produces no infinitesimal area distortion at the south pole.
Other authors use a sphere of radius and the plane . In this case the formulae become
In general, one can define a stereographic projection from any point on the sphere onto any plane such that
* is perpendicular to the diameter through , and
* does not contain .
As long as meets these conditions, then for any point other than the line through and meets in exactly one point , which is defined to be the stereographic projection of P onto E. | 1 | Crystallography |
Transformations are often seen to follow a characteristic s-shaped, or sigmoidal, profile where the transformation rates are low at the beginning and the end of the transformation but rapid in between.
The initial slow rate can be attributed to the time required for a significant number of nuclei of the new phase to form and begin growing. During the intermediate period the transformation is rapid as the nuclei grow into particles and consume the old phase while nuclei continue to form in the remaining parent phase.
Once the transformation approaches completion, there remains little untransformed material for further nucleation, and the production of new particles begins to slow. Additionally, the previously formed particles begin to touch one another, forming a boundary where growth stops. | 1 | Crystallography |
Thermometric titrimetry offers a rapid, highly precise method for the determination of aluminium in solution. A solution of aluminium is conditioned with acetate buffer and an excess of sodium and potassium ions. Titration with sodium or potassium fluoride yields the exothermic precipitation of an insoluble alumino-fluoride salt.
: Al + Na + 2K + 6F ↔ KNaAlF↓
Because 6 mole of fluoride react with one mole of aluminium, the titration is particularly precise, and a coefficient of variance (CV) of 0.03 has been achieved in the analysis of alum.
When aluminium ion (say as aluminium nitrate) is employed as the titrant, fluoride can be determined using the same chemistry. This titration is useful in the determination of fluoride in complex acid mixtures used as etchants in the semi-conductor industry. | 0 | Chromatography + Titration + pH indicators |
Twin laws are symmetry operations that define the orientation between twin crystal segments. These are as characteristic of the mineral as are its crystal face angles. For example, crystals of staurolite show twinning at angles of almost precisely 90 degrees or 30 degrees. A twin law is not a symmetry operation of the full set of basis points.
Twin laws include reflection operations, rotation operations, and the inversion operation. Reflection twinning is described by the Miller indices of the twin plane (i.e. {hkl}) while rotational twinning is described by the direction of the twin axis (i.e. <hkl>). Inversion twinning is typically equivalent to a reflection or rotation symmetry.
Rotational twin laws are almost always 2-fold rotations, though any other permitted rotation symmetry (3-fold, 4-fold, 5-fold or 6-fold) is possible. The twin axis will be perpendicular to a lattice plane. It is possible for a rotational twin law to share the same axis as a rotational symmetry of the individual crystal if the twin law is a 2-fold rotation and the symmetry operation is a 3-fold rotation. This is the case for spinel law twinning on <111>: The spinel structure has a 3-fold rotational symmetry on <111> and spinel is commonly twinned by 2-fold rotation on <111>.
The boundary between crystal segments is called a composition surface or, if it is planar, a composition plane. The composition plane is often, though not always, parallel to the twin law plane of a reflection law. If this is the case, the twin plane is always parallel to a possible crystal face. | 1 | Crystallography |
Cortisone acetate exists in at least five different polymorphs, four of which are unstable in water and change to a stable form. | 1 | Crystallography |
To the left is the NFPA diamond as determined by the Safety Data Sheet, or SDS, by Fisher Scientific. There is minimal risk in handling the chemical. | 0 | Chromatography + Titration + pH indicators |
The electron gun is one of the most important piece of equipment in a RHEED system. The gun limits the resolution and testing limits of the system. Tungsten filaments are the primary electron source for the electron gun of most RHEED systems due to the low work function of tungsten. In the typical setup, the tungsten filament is the cathode and a positively biased anode draws electrons from the tip of the tungsten filament.
The magnitude of the anode bias determines the energy of the incident electrons. The optimal anode bias is dependent upon the type of information desired. At large incident angles, electrons with high energy can penetrate the surface of the sample and degrade the surface sensitivity of the instrument. However, the dimensions of the Laue zones are proportional to the inverse square of the electron energy meaning that more information is recorded at the detector at higher incident electron energies. For general surface characterization, the electron gun is operated the range of 10-30 keV.
In a typical RHEED setup, one magnetic and one electric field focus the incident beam of electrons. A negatively biased Wehnelt electrode positioned between the cathode filament and anode applies a small electric field, which focuses the electrons as they pass through the anode. An adjustable magnetic lens focuses the electrons onto the sample surface after they pass through the anode. A typical RHEED source has a focal length around 50 cm. The beam is focused to the smallest possible point at the detector rather than the sample surface so that the diffraction pattern has the best resolution.
Phosphor screens that exhibit photoluminescence are widely used as detectors. These detectors emit green light from areas where electrons hit their surface and are common to TEM as well. The detector screen is useful for aligning the pattern to an optimal position and intensity. CCD cameras capture the patterns to allow for digital analysis. | 1 | Crystallography |
GPC is a type of chromatography in which analytes are separated, based on their size or hydrodynamic volume (radius of gyration). This differs from other chromatographic techniques, which depend upon chemical or physical interactions between the mobile and stationary phases to separate analytes. Separation occurs via the use of porous gel beads packed inside a column (see stationary phase (chemistry)). The principle of separation relies on the differential exclusion or inclusion of the macromolecules by the porous gel stationary phase. Larger molecules are excluded from entering the pores and elute earlier, while smaller molecules can enter the pores, thus staying longer inside the column. The entire process takes place without any interaction of the analytes with the surface of the stationary phase. The smaller analytes relative to the pore sizes can permeate these pores and spend more time inside the gel particles, increasing their retention time. Conversely, larger analytes relative to the pores sizes spend little if any time inside the column, hence they elute sooner. Each type of column has a range of molecular weights that can be separated, according to their pores sizes. If an analyte is too large relative to the column's pores, it will not be retained at all and will be totally excluded; conversely, if the analyte is small relative to the pores sizes, it will be totally permeating. Analytes that are totally excluded, elute with the free volume outside around the particles (V), the total exclusion limit, while analytes that are completely delayed, elute with the solvent, marking the total permeation volume of the column, including also the solvent held inside the pores (V). The total volume can be considered by the following equation, where V is the volume of the polymer gel and V is the total volume:
As can be inferred, there is a limited range of molecular weights that can be separated by each column, therefore the size of the pores for the packing should be chosen according to the range of molecular weight of analytes to be separated. For polymer separations the pore sizes should be on the order of the polymers being analyzed. If a sample has a broad molecular weight range it may be necessary to use several GPC columns with varying pores volumes in tandem to resolve the sample fully. | 0 | Chromatography + Titration + pH indicators |
In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogist Charles-Victor Mauguin (who modified it in 1931). This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.
The Hermann–Mauguin notation, compared with the Schoenflies notation, is preferred in crystallography because it can easily be used to include translational symmetry elements, and it specifies the directions of the symmetry axes. | 1 | Crystallography |
In metal borides, the bonding of boron varies depending on the atomic ratio B/M. Diborides have B/M = 2, as in the well-known superconductor MgB; they crystallize in a hexagonal AlB-type layered structure. Hexaborides have B/M = 6 and form a three-dimensional boron framework based on a boron octahedron (Fig. 1a). Tetraborides, i.e. B/M = 4, are mixtures of diboride and hexaboride structures. Cuboctahedron (Fig. 1b) is the structural unit of dodecaborides, which have a cubic lattice and B/M = 12. When the composition ratio exceeds 12, boron forms B icosahedra (Fig. 1c) which are linked into a three-dimensional boron framework, and the metal atoms reside in the voids of this framework.
This complex bonding behavior originates from the fact that boron has only three valence electrons; this hinders tetrahedral bonding as in diamond or hexagonal bonding as in graphite. Instead, boron atoms form polyhedra. For example, three boron atoms make up a triangle where they share two electrons to complete the so-called three-center bonding. Boron polyhedra, such as B octahedron, B cuboctahedron and B icosahedron, lack two valence electrons per polyhedron to complete the polyhedron-based framework structure. Metal atoms need to donate two electrons per boron polyhedron to form boron-rich metal borides. Thus, boron compounds are often regarded as electron-deficient solids. The covalent bonding nature of metal boride compounds also give them their hardness and inert chemical reactivity property.
Icosahedral B compounds include α-rhombohedral boron (BC), β-rhombohedral boron (MeB, 23≤x), α-tetragonal boron (BBC), β-tetragonal boron (β-AlB), AlB or AlCB, YB, YB, YB, NaB or MgAlB, γ-AlB, BeB and SiB.
YB and YB decompose without melting that hinders their growth as single crystals by the floating zone method. However, addition of a small amount of Si solves this problem and results in single crystals with the stoichiometry of YBSi. This stabilization technique allowed the synthesis of some other boron-rich rare-earth borides.
Albert and Hillebrecht reviewed binary and selected ternary boron compounds containing main-group elements, namely, borides of the alkali and alkaline-earth metals, aluminum borides and compounds of boron and the nonmetals C, Si, Ge, N, P, As, O, S and Se. They, however, excluded the described here icosahedron-based rare-earth borides. Note that rare-earth elements have d- and f-electrons that complicates chemical and physical properties of their borides. Werheit et al. reviewed Raman spectra of numerous icosahedron-based boron compounds.
Figure 2 shows a relationship between the ionic radius of trivalent rare-earth ions and the composition of some rare-earth borides. Note that scandium has many unique boron compounds, as shown in figure 2, because of the much smaller ionic radius compared with other rare-earth elements.
In understanding the crystal structures of rare-earth borides, it is important to keep in mind the concept of partial site occupancy, that is, some atoms in the described below unit cells can take several possible positions with a given statistical probability. Thus, with the given statistical probability, some of the partial-occupancy sites in such a unit cell are empty, and the remained sites are occupied. | 1 | Crystallography |
Forestal is a solvent used in chromatography, composed of acetic acid, water, and hydrochloric acid in a 30:10:3 ratio by volume. It is useful for isolating anthocyanins in room-temperature chromatography using standard filter paper. | 0 | Chromatography + Titration + pH indicators |
In some transitions a number of atoms occupying crystallographic positions that were originally equivalent will move away slightly from their ideal positions according to a certain pattern. This pattern or repeat motif may span multiple unit cells. The cause of this phenomenon is the small changes in chemical bonding that favor formations of semi-regular and larger clusters of atoms. Although having the undistorted substructure, these materials are typically unsaturated in the sense that one of the bands in the band structure is only partially filled. The distortion changes the band structure, in part splitting the bands up into smaller bands that can be more completely filled or emptied to lower the energy of the system. This process may not go to completion, however, because the substructure only allows for a certain amount of distortion. Superstructures of this type are often incommensurate. A good example is found in the structural transitions of 1T-TaS, a compound with a partially filled, narrow d band (Ta(IV) has a d configuration). | 1 | Crystallography |
The Urbach tail is an exponential part in the energy spectrum of the absorption coefficient. This tail appears near the optical band edge in amorphous, disordered and crystalline materials. | 1 | Crystallography |
In crystallography, the orientations of crystal axes and faces in three-dimensional space are a central geometric concern, for example in the interpretation of X-ray and electron diffraction patterns. These orientations can be visualized as in the section Visualization of lines and planes above. That is, crystal axes and poles to crystal planes are intersected with the northern hemisphere and then plotted using stereographic projection. A plot of poles is called a pole figure.
In electron diffraction, Kikuchi line pairs appear as bands decorating the intersection between lattice plane traces and the Ewald sphere thus providing experimental access to a crystal's stereographic projection. Model Kikuchi maps in reciprocal space, and fringe visibility maps for use with bend contours in direct space, thus act as road maps for exploring orientation space with crystals in the transmission electron microscope. | 1 | Crystallography |
In the pharmaceutical industry, acid-base titration serves as a fundamental analytical technique with diverse applications. One primary use involves the determination of the concentration of Active Pharmaceutical Ingredients (APIs) in drug formulations, ensuring product quality and compliance with regulatory standards.
Acid–base titration is particularly valuable in quantifying acidic or basic functional groups with pharmaceutical compounds. Additionally, the method is employed for the analysis of additives or ingredients, making it easier to adjust and control how a product is made. Quality control laboratories utilize acid-base titration to assess the purity of raw materials and to monitor various stages of drug manufacturing processes.
The technique's reliability and simplicity make it an integral tool in pharmaceutical research and development, contributing to the production of safe and effective medications. | 0 | Chromatography + Titration + pH indicators |
In chromatography, the is defined as the ratio of the distance traveled by the center of a spot to the distance traveled by the solvent front. Ideally, the values for R are equivalent to the R values used in column chromatography.
Although the term retention factor is sometimes used synonymously with retardation factor in regard to planar chromatography the term is not defined in this context. However, in column chromatography, the retention factor or capacity factor (k) is defined as the ratio of time an analyte is retained in the stationary phase to the time it is retained in the mobile phase, which is inversely proportional to the retardation factor. | 0 | Chromatography + Titration + pH indicators |
The stereographic is the only projection that maps all circles on a sphere to circles on a plane. This property is valuable in planetary mapping where craters are typical features. The set of circles passing through the point of projection have unbounded radius, and therefore degenerate into lines. | 1 | Crystallography |
In the Mohr method, named after Karl Friedrich Mohr, potassium chromate is an indicator, giving red silver chromate after all chloride ions have reacted:
: 2Ag (aq) + CrO (aq) → AgCrO (s) (K = 1.1 × 10)
The solution needs to be near neutral, because silver hydroxide forms at high pH, while the chromate forms AgCrO or AgHCrO4 at low pH, reducing the concentration of chromate ions, and delaying the formation of the precipitate. Carbonates and phosphates precipitate with silver, and need to be absent to prevent inaccurate results.
The Mohr method may be adapted to determine the total chlorine content of a sample by igniting the sample with calcium, then ferric acetate. Calcium acetate "fixes" free chlorine, precipitates carbonates, and neutralizes the resultant solution. Ferric acetate removes phosphates. All chlorides are dissolved out of the residue, and titrated. | 0 | Chromatography + Titration + pH indicators |
Some other lattice packings are often found in physical systems. These include the cubic lattice with a density of , the hexagonal lattice with a density of and the tetrahedral lattice with a density of . | 1 | Crystallography |
The FCC and HCP packings are the densest known packings of equal spheres with the highest symmetry (smallest repeat units).
Denser sphere packings are known, but they involve unequal sphere packing.
A packing density of 1, filling space completely, requires non-spherical shapes, such as honeycombs.
Replacing each contact point between two spheres with an edge connecting the centers of the touching spheres produces tetrahedrons and octahedrons of equal edge lengths.
The FCC arrangement produces the tetrahedral-octahedral honeycomb.
The HCP arrangement produces the gyrated tetrahedral-octahedral honeycomb.
If, instead, every sphere is augmented with the points in space that are closer to it than to any other sphere, the duals of these honeycombs are produced: the rhombic dodecahedral honeycomb for FCC, and the trapezo-rhombic dodecahedral honeycomb for HCP.
Spherical bubbles appear in soapy water in a FCC or HCP arrangement when the water in the gaps between the bubbles drains out. This pattern also approaches the rhombic dodecahedral honeycomb or trapezo-rhombic dodecahedral honeycomb. However, such FCC or HCP foams of very small liquid content are unstable, as they do not satisfy Plateau's laws. The Kelvin foam and the Weaire–Phelan foam are more stable, having smaller interfacial energy in the limit of a very small liquid content.
There are two types of interstitial holes left by hcp and fcc conformations; tetrahedral and octahedral void. Four spheres surround the tetrahedral hole with three spheres being in one layer and one sphere from the next layer. Six spheres surround an octahedral voids with three spheres coming from one layer and three spheres coming from the next layer. Structures of many simple chemical compounds, for instance, are often described in terms of small atoms occupying tetrahedral or octahedral holes in closed-packed systems that are formed from larger atoms.
Layered structures are formed by alternating empty and filled octahedral planes. Two octahedral layers usually allow for four structural arrangements that can either be filled by an hpc of fcc packing systems. In filling tetrahedral holes a complete filling leads to fcc field array. In unit cells, hole filling can sometimes lead to polyhedral arrays with a mix of hcp and fcc layering. | 1 | Crystallography |
Deformation twinning is a response to shear stress. The crystal structure is displaced along successive planes of the crystal, a process also called glide. The twinning is always reflection twinning and the glide plane is also the mirror plane. Deformation twinning can be observed in a calcite cleavage fragment by applying gentle pressure with a knife blade near an edge. This particular glide twinning, {102}, is found almost universally in deformed rock beds containing calcite.
Twinning and slip are competitive mechanisms for crystal deformation. Each mechanism is dominant in certain crystal systems and under certain conditions. In fcc metals, slip is almost always dominant because the stress required is far less than twinning stress.
Twinning can occur by cooperative displacement of atoms along the face of the twin boundary. This displacement of a large quantity of atoms simultaneously requires significant energy to perform. Therefore, the theoretical stress required to form a twin is quite high. It is believed that twinning is associated with dislocation motion on a coordinated scale, in contrast to slip, which is caused by independent glide at several locations in the crystal.
Compared to slip, twinning produces a deformation pattern that is more heterogeneous in nature. This deformation produces a local gradient across the material and near intersections between twins and grain boundaries. The deformation gradient can lead to fracture along the boundaries, particularly in bcc transition metals at low temperatures.
Of the three common crystalline structures bcc, fcc, and hcp, the hcp structure is the most likely to form deformation twins when strained, because they rarely have a sufficient number of slip systems for an arbitrary shape change. High strain rates, low stacking-fault energy and low temperatures facilitate deformation twinning.
If a metal with face-centered cubic (fcc) structure, like Al, Cu, Ag, Au, etc., is subjected to stress, it will experience twinning. The formation and migration of twin boundaries is partly responsible for ductility and malleability of fcc metals.
Twin boundaries are partly responsible for shock hardening and for many of the changes that occur in cold work of metals with limited slip systems or at very low temperatures. They also occur due to martensitic transformations: the motion of twin boundaries is responsible for the pseudoelastic and shape-memory behavior of nitinol, and their presence is partly responsible for the hardness due to quenching of steel. In certain types of high strength steels, very fine deformation twins act as primary obstacles against dislocation motion. These steels are referred to as TWIP steels, where TWIP stands for twinning-induced plasticity. | 1 | Crystallography |
In three-dimensional space there are 14 Bravais lattices. These are obtained by combining one of the seven lattice systems with one of the centering types. The centering types identify the locations of the lattice points in the unit cell as follows:
*Primitive (P): lattice points on the cell corners only (sometimes called simple)
*Base-centered (S: A, B, or C): lattice points on the cell corners with one additional point at the center of each face of one pair of parallel faces of the cell (sometimes called end-centered)
*Body-centered (I): lattice points on the cell corners, with one additional point at the center of the cell
*Face-centered (F): lattice points on the cell corners, with one additional point at the center of each of the faces of the cell
Not all combinations of lattice systems and centering types are needed to describe all of the possible lattices, as it can be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below. Below each diagram is the Pearson symbol for that Bravais lattice.
Note: In the unit cell diagrams in the following table all the lattice points on the cell boundary (corners and faces) are shown; however, not all of these lattice points technically belong to the given unit cell. This can be seen by imagining moving the unit cell slightly in the negative direction of each axis while keeping the lattice points fixed. Roughly speaking, this can be thought of as moving the unit cell slightly left, slightly down, and slightly out of the screen. This shows that only one of the eight corner lattice points (specifically the front, left, bottom one) belongs to the given unit cell (the other seven lattice points belong to adjacent unit cells). In addition, only one of the two lattice points shown on the top and bottom face in the Base-centered column belongs to the given unit cell. Finally, only three of the six lattice points on the faces in the Face-centered column belongs to the given unit cell.
The unit cells are specified according to six lattice parameters which are the relative lengths of the cell edges (a, b, c) and the angles between them (α, β, γ), where α is the angle between b and c, β is the angle between a and c, and γ is the angle between a and b. The volume of the unit cell can be calculated by evaluating the triple product , where a, b, and c are the lattice vectors. The properties of the lattice systems are given below:
Some basic information for the lattice systems and Bravais lattices in three dimensions is summarized in the diagram at the beginning of this page. The seven sided polygon (heptagon) and the number 7 at the centre indicate the seven lattice systems. The inner heptagons indicate the lattice angles, lattice parameters, Bravais lattices and Schöenflies notations for the respective lattice systems. | 1 | Crystallography |
The general mathematical concept embodied in a Wigner–Seitz cell is more commonly called a Voronoi cell, and the partition of the plane into these cells for a given set of point sites is known as a Voronoi diagram.
The cell may be chosen by first picking a lattice point. After a point is chosen, lines are drawn to all nearby lattice points. At the midpoint of each line, another line is drawn normal to each of the first set of lines. The smallest area enclosed in this way is called the Wigner–Seitz primitive cell.
For a 3-dimensional lattice, the steps are analogous, but in step 2 instead of drawing perpendicular lines, perpendicular planes are drawn at the midpoint of the lines between the lattice points.
As in the case of all primitive cells, all area or space within the lattice can be filled by Wigner–Seitz cells and there will be no gaps.
Nearby lattice points are continually examined until the area or volume enclosed is the correct area or volume for a primitive cell. Alternatively, if the basis vectors of the lattice are reduced using lattice reduction only a set number of lattice points need to be used. In two-dimensions only the lattice points that make up the 4 unit cells that share a vertex with the origin need to be used. In three-dimensions only the lattice points that make up the 8 unit cells that share a vertex with the origin need to be used. | 1 | Crystallography |
The A. V. Shubnikov Institute of Crystallography is a scientific institute of the Department of Physical Sciences of the Russian Academy of Sciences (RAS) located in Moscow, Russia. The institute was created by the order of the Presidium of the Academy of Sciences of the USSR on 16November 1943. The first director of the Institute was a corresponding member of the Academy of Sciences of the USSR Alexei Vasilievich Shubnikov. In 1969, the institute was awarded the Order of the Red Banner of Labour.
Areas of scientific interest:
* Crystal growth: research into crystal formation and growth, development of synthesis methods and creation of equipment for crystallography
* Crystal structure: study of idealialized (theoretical) and real-world crystal structures
* Crystal properties: study of symmetry and physical properties of crystals; search for crystals with valuable properties | 1 | Crystallography |
Bromothymol blue (also known as bromothymol sulfone phthalein and BTB) is a pH indicator. It is mostly used in applications that require measuring substances that would have a relatively neutral pH (near 7). A common use is for measuring the presence of carbonic acid in a liquid. It is typically sold in solid form as the sodium salt of the acid indicator. | 0 | Chromatography + Titration + pH indicators |
Kennedy became a professor of chemistry at the University of Florida in 1991. After 11 years, he moved to the University of Michigan. He has graduated approximately 80 graduate students. Kennedy’s research focuses on developing analytical instrumentation and methods that can help solve biological problems. He is considered a leader in the field of analytical chemistry, and an expert in endocrinology, neurochemistry, and high-throughput analysis. Major contributions to analytical chemistry include affinity probe capillary electrophoresis, in vivo neurochemical measurements, and ultra-high pressure liquid chromatography. He has been a Lilly Analytical Research Fellow, Alfred P. Sloan Fellow, NSF Presidential Faculty Fellow, and AAAS Fellow. | 0 | Chromatography + Titration + pH indicators |
Streak seeding is a method first described during ICCBM-3 by Enrico Stura to induce crystallization in a straight line into a sitting or hanging drop for protein crystallization by introducing microseeds. The purpose is to control nucleation and understand the parameters that make crystals grow. It is also used to test any particular set of conditions to check if crystals could grow under such conditions.
The technique is relatively simple. A cat whisker is used to dislodge seeds from a crystal. The whisker is passed through the drop starting from one side of the drop and ending on the opposite side of the drop in one smooth motion. To allow for vapour diffusion equilibration, the well in which the drop has been placed is resealed. The same procedure is repeated for all the drops whose conditions need testing. | 1 | Crystallography |
The diamond cubic crystal structure occurs for example diamond (carbon), tin, and most semiconductors. There are 8 atoms in the cubic unit cell. We can consider the structure as a simple cubic with a basis of 8 atoms, at positions
But comparing this to the FCC above, we see that it is simpler to describe the structure as FCC with a basis of two atoms at (0, 0, 0) and (1/4, 1/4, 1/4). For this basis, Equation () becomes:
And then the structure factor for the diamond cubic structure is the product of this and the structure factor for FCC above, (only including the atomic form factor once)
with the result
* If h, k, ℓ are of mixed parity (odd and even values combined) the first (FCC) term is zero, so
* If h, k, ℓ are all even or all odd then the first (FCC) term is 4
** if h+k+ℓ is odd then
** if h+k+ℓ is even and exactly divisible by 4 () then
** if h+k+ℓ is even but not exactly divisible by 4 () the second term is zero and
These points are encapsulated by the following equations:
where is an integer. | 1 | Crystallography |
Chromatographic peak resolution is given by
where t is the retention time and w is the peak width at baseline. Here compound 1 elutes before compound 2.
If the peaks have the same width | 0 | Chromatography + Titration + pH indicators |
Perovskite materials exhibit many interesting and intriguing properties from both the theoretical and the application point of view. Colossal magnetoresistance, ferroelectricity, superconductivity, charge ordering, spin dependent transport, high thermopower and the interplay of structural, magnetic and transport properties are commonly observed features in this family. These compounds are used as sensors and catalyst electrodes in certain types of fuel cells and are candidates for memory devices and spintronics applications.
Many superconducting ceramic materials (the high temperature superconductors) have perovskite-like structures, often with 3 or more metals including copper, and some oxygen positions left vacant. One prime example is yttrium barium copper oxide which can be insulating or superconducting depending on the oxygen content.
Chemical engineers are considering a cobalt-based perovskite material as a replacement for platinum in catalytic converters for diesel vehicles. | 1 | Crystallography |
In the thermometric titration, titrant is added at a known constant rate to a titrand until the completion of the reaction is indicated by a change in temperature. The endpoint is determined by an inflection in the curve generated by the output of a temperature measuring device.
Consider the titration reaction:
: aA + bB = pP (3)
Where:
: A = the titrant, and a = the corresponding number of moles reacting
: B = the analyte, and b = the corresponding number of moles reacting
: P = the product, and p = the corresponding number of moles produced
At completion, the reaction produces a molar heat of reaction ΔH which is shown as a measurable temperature change ΔT. In an ideal system, where no losses or gains of heat due to environmental influences are involved, the progress of the reaction is observed as a constant increase or decrease of temperature depending respectively on whether ΔH is negative (indicating an exothermic reaction) or positive (indicating an endothermic reaction). In this context, environmental influences may include (in order of importance):
*Heat losses or gains from outside the system via the vessel walls and cover;
*Differences in the temperature between the titrant and the titrand;
*Evaporative losses from the surface of the rapidly mixed fluid;
*Heats of solution when the titrant solvent is mixed with the analyte solvent;
*Heat introduced by the mechanical action of stirring (minor influence); and
*Heat produced by the thermistor itself (very minor influence).
If the equilibrium for the reaction lies far to the right (i.e. a stoichiometric equilibrium has been achieved), then when all analyte has been reacted by the titrant continuing addition of titrant will be revealed by a sharp break in the temperature/volume curve. Figures 1a and 1b illustrate idealized examples.
The shape of experimentally obtained thermometric titration plots will vary from such idealized examples, and some of the environmental influences listed above may have impacts. Curvature at the endpoint might be observed. This can be due to insensitivity of the sensor or where thermal equilibrium at the endpoint is slow to occur. It can also occur where the reaction between titrant and titrand does not proceed to stoichiometric completion. The determinant of the degree to which a reaction will proceed to completion is the free energy change. If this is favourable, then the reaction will proceed to be completion and be essentially stoichiometric. In this case, the sharpness of the endpoint is dependent on the magnitude of the enthalpy change. If it is unfavourable, the endpoint will be rounded regardless of the magnitude of the enthalpy change. Reactions where non-stoichiometric equilibria are evident can be used to obtain satisfactory results using a thermometric titration approach. If the portions of the titration curve both prior to and after the endpoint are reasonably linear, then the intersection of tangents to these lines will accurately locate the endpoint. This is illustrated in Figure 2.
Consider the reaction for the equation aA + bB = pP which is non-stoichiometric at equilibrium. Let A represent the titrant, and B the titrand. At the beginning of the titration, the titrand B is strongly in excess, and the reaction is pushed towards completion. Under these conditions, for a constant rate of titrant addition the temperature increase is constant and the curve is essentially linear until the endpoint is approached. In a similar manner, when the titrant is in excess past the endpoint, a linear temperature response can also be anticipated. Thus intersection of tangents will reveal the true endpoint.
An actual thermometric titration plot for the determination of a strong base with a strong acid is illustrated in Figure 3.
The most practical sensor for measuring temperature change in titrating solutions has been found to be the thermistor. Thermistors are small solid state devices which exhibit relatively large changes in electrical resistance for small changes in temperature. They are manufactured from sintered mixed metal oxides, with lead wires enabling connection to electrical circuitry. The thermistor is encapsulated in a suitable electrically insulating medium with satisfactory heat transfer characteristics and acceptable chemical resistance. Typically for thermistors used for chemical analysis the encapsulating medium is glass, although thermistors encapsulated in epoxy resin may be used in circumstances where either chemical attack (e.g., by acidic fluoride-containing solutions) or severe mechanical stress is anticipated. The thermistor is supported by suitable electronic circuitry to maximize sensitivity to minute changes in solution temperature. The circuitry in the Metrohm 859 Titrotherm thermometric titration interface module is capable of resolving temperature changes as low as 10 K.
A critical element in modern automated thermometric titrimetry is the ability to locate the endpoint with a high degree of reproducibility. It is clearly impractical and insufficient for modern demands of accuracy and precision to estimate the inflection by intersection of tangents. This is done conveniently by derivatization of the temperature curve. The second derivative essentially locates the intersection of tangents to the temperature curve immediately pre- and post- the breakpoint.
Thermistors respond quickly to small changes in temperature such as temperature gradients in the mixed titration solution, and thus the signal can exhibit a small amount of noise. Prior to derivatization it is therefore necessary to digitally smooth (or "filter") the temperature curve in order to obtain sharp, symmetrical second derivative "peaks" which will accurately locate the correct inflection point. This is illustrated in Figure 5. The degree of digital smoothing is optimized for each determination, and is stored as a method parameter for application every time a titration for that particular analysis is run.
Because enthalpy change is a universal characteristic of chemical reactions, thermometric endpoint sensing can be applied to a wide range of titration types, e.g.
* Acid/base
* Redox
* Complexometric (EDTA) and
* Precipitation
Further, since the sensor is not required to interact with the titration solution electrochemically, titrations in non-conducting media can be performed, as can titrations using reactions for which no convenient or cost-effective potentiometric sensor is available.
Thermometric titrations generally demand rapid reaction kinetics in order to obtain sharp reproducible endpoints. Where reaction kinetics are slow, and direct titrations between titrant and titrand are not possible, indirect or back-titrations often can be devised to solve the problem.
Catalytically enhanced endpoints can be used in some instances where the temperature change at the endpoint is very small and endpoints would not be detected satisfactorily by the titration software.
The suitability of a particular chemical reaction as a candidate for a thermometric titration procedure can generally be predicted on the basis of the estimated amount of analyte present in the sample and the enthalpy of the reaction. However, other parameters such as the kinetics of the reaction, the sample matrix itself, heats of dilution and losses of heat to the environment can affect the outcome. A properly designed experimental program is the most reliable way of determining the viability of a thermometric titration approach. Successful applications for thermometric titrations are generally where titrant-titrand reaction kinetics are fast, and chemical equilibria are stoichiometric or nearly so. | 0 | Chromatography + Titration + pH indicators |
For a finite crystal means that the sums in equations 1-7 are now over a finite . The effect is most easily demonstrated with a 1-D lattice of points. The sum of the phase factors is a geometric series and the structure factor becomes:
This function is shown in the Figure for different values of .
When the scattering from every particle is in phase, which is when the scattering is at a reciprocal lattice point , the sum of the amplitudes must be and so the maxima in intensity are . Taking the above expression for and estimating the limit using, for instance, LHôpitals rule) shows that as seen in the Figure. At the midpoint (by direct evaluation) and the peak width decreases like . In the large limit, the peaks become infinitely sharp Dirac delta functions, the reciprocal lattice of the perfect 1-D lattice.
In crystallography when is used, is large, and the formal size effect on diffraction is taken as , which is the same as the expression for above near to the reciprocal lattice points, . Using convolution, we can describe the finite real crystal structure as [lattice] [basis] rectangular function, where the rectangular function has a value 1 inside the crystal and 0 outside it. Then [crystal structure] = [lattice] [basis] [rectangular function]; that is, scattering [reciprocal lattice] [structure factor] [ sinc function]. Thus the intensity, which is a delta function of position for the perfect crystal, becomes a function around every point with a maximum , a width , area . | 1 | Crystallography |
Some pairs of minerals that are not related structurally or compositionally may also exhibit epitaxy. A common example is rutile TiO on hematite FeO. Rutile is tetragonal and hematite is trigonal, but there are directions of similar spacing between the atoms in the (100) plane of rutile (perpendicular to the a axis) and the (001) plane of hematite (perpendicular to the c axis). In epitaxy these directions tend to line up with each other, resulting in the axis of the rutile overgrowth being parallel to the c axis of hematite, and the c axis of rutile being parallel to one of the axes of hematite. | 1 | Crystallography |
When one atom substitutes for one of the principal atomic components within the crystal structure, alteration in the electrical and thermal properties of the material may ensue. Impurities may also manifest as electron spin impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of certain properties such as specific heat may be affected by small concentrations of an impurity, as for example impurities in semiconducting ferromagnetic alloys may lead to different properties as first predicted in the late 1960s. | 1 | Crystallography |
The flow cells are connected to a display and/or recorder. On older systems this was a simple chart recorder, on modern systems a computer with hardware interface and display is used. This permits the experimenter to identify when peaks in protein concentration occur, indicating that specific components of the mixture are being eluted. | 0 | Chromatography + Titration + pH indicators |
Friedel's salt could be first tentatively represented as an AFm phase in which two chloride ions would have simply replaced one sulfate ion. This conceptual representation based on the intuition of a simple stoichiometric exchange is very convenient to remind but such a simple mechanism likely does not directly occur and must be considered with caution:
Indeed, the reality appears to be more complex than such a simple stoichiometric exchange between chloride and sulfate ions in the AFm crystal structure. In fact, it seems that chloride ions are electrostatically sorbed onto the positively charged [CaAl(OH) · 2HO] layer of AFm hydrate, or could also exchange with hydroxide ions (OH) also present in the interlayer. So, the simple and "apparent" exchange reaction first presented here above for the sake of ease does not correspond to the reality and is an oversimplified representation.
Similarly, Kuzel’s salt could seem to be formed when only 1 Cl ion exchanges with in AFm (half substitution of sulfate ions):
Glasser et al. (1999) proposed to name this half-substituted salt in honor of his discoverer: Hans-Jürgen Kuzel.
However, Mesbah et al. (2011) have identified two different types of interlayers in the crystallographic structure they have determined and it precludes the common anion exchange reaction presented here above as stated by the authors themselves in their conclusions:
So, if the global chemical composition of Friedels salt and Kuzels salt corresponds well respectively with the stoichiometry of a complete substitution, or a half substitution, of sulfate ions by chloride ions in the crystal structure of AFm, it does not tell directly anything on the exact mechanism of anion substitution in this complicated system. Only detailed and well controlled chloride sorption, or anion exchange, experiments with a complete analysis of all the dissolved species present in aqueous solution (also including OH, Na and Ca ions) can decipher the system. | 1 | Crystallography |
Theoretical descriptions of contrast formation in X-ray topography are largely based on the dynamical theory of diffraction. This framework is helpful in the description of many aspects of topographic image formation: entrance of an X-ray wavefield into a crystal, propagation of the wavefield inside the crystal, interaction of wavefield with crystal defects, altering of wavefield propagation by local lattice strains, diffraction, multiple scattering, absorption.
The theory is therefore often helpful in the interpretation of topographic images of crystal defects. The exact nature of a defect often cannot be deduced directly from the observed image (i.e., a "backwards calculation" is problematic). Instead, one has to make assumptions about the structure of the defect, deduce a hypothetical image from the assumed structure ("forward calculation", based on theory), and compare with the experimental image. If the match between both is not good enough, the assumptions have to be varied until sufficient correspondence is reached. Theoretical calculations, and in particular numerical simulations by computer based on this theory, are thus a valuable tool for the interpretation of topographic images. | 1 | Crystallography |
A body-centered cubic unit cell has six octahedral voids located at the center of each face of the unit cell, and twelve further ones located at the midpoint of each edge of the same cell, for a total of six net octahedral voids. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void, for a total of twelve net tetrahedral voids. These tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. | 1 | Crystallography |
The International Committee of the Red Cross (ICRC) uses forensic science for humanitarian purposes to clarify the fate of missing persons after armed conflict, disasters or migration, and is one of the services related to Restoring Family Links and Missing Persons. Knowing what has happened to a missing relative can often make it easier to proceed with the grieving process and move on with life for families of missing persons.
Forensic science is used by various other organizations to clarify the fate and whereabouts of persons who have gone missing. Examples include the NGO Argentine Forensic Anthropology Team, working to clarify the fate of people who disappeared during the period of the 1976–1983 military dictatorship. The International Commission on Missing Persons (ICMP) used forensic science to find missing persons, for example after the conflicts in the Balkans.
Recognising the role of forensic science for humanitarian purposes, as well as the importance of forensic investigations in fulfilling the state's responsibilities to investigate human rights violations, a group of experts in the late-1980s devised a UN Manual on the Prevention and Investigation of Extra-Legal, Arbitrary and Summary Executions, which became known as the Minnesota Protocol. This document was revised and re-published by the Office of the High Commissioner for Human Rights in 2016. | 0 | Chromatography + Titration + pH indicators |
Because digitized 2D periodic images are in the information theoretical approach just data organized in 2D arrays of pixels, core features of Crystallographic Image Processing can be utilized independent of the type of microscope with which the images/data were recorded. The CIP technique has, accordingly been applied (on the basis of the 2dx program) to atomic resolution Z-contrast images of Si-clathrates, as recorded in an aberration-corrected scanning transmission electron microscope. Images of 2D periodic arrays of flat lying molecules on a substrate as recorded with scanning tunneling microscopes were also crystallographic processed utilizing the program CRISP. | 1 | Crystallography |
Erythrolitmin (also called erythrolein) is the active ingredient extracted from the Litmus lichen, used in chemistry as a pH indicator. Erythrolitmin is related to the orceins, and consists essentially of several phenoxazone and orcinol residues. | 0 | Chromatography + Titration + pH indicators |
Solutions to wave propagation problems in linear elastic transversely isotropic media can be constructed by superposing solutions for the quasi-P wave, the quasi S-wave, and a S-wave polarized orthogonal to the quasi S-wave.
However, the equations for the angular variation of velocity are algebraically complex and the plane-wave velocities are functions of the propagation angle are. The direction dependent wave speeds for elastic waves through the material can be found by using the Christoffel equation and are given by
where is the angle between the axis of symmetry and the wave propagation direction, is mass density and the are elements of the elastic stiffness matrix. The Thomsen parameters are used to simplify these expressions and make them easier to understand. | 1 | Crystallography |
The Ripper Method is commonly used in wine making applications as SO is often added to wine to maintain its freshness and the concentration needs to be determined. The technique is not precise and is prone to systematic error as well. This limits its use, despite being a fast and inexpensive test. | 0 | Chromatography + Titration + pH indicators |
The word "litmus" comes from an Old Norse word for “moss used for dyeing”. About 1300, the Spanish physician Arnaldus de Villa Nova began using litmus to study acids and bases.
From the 16th century onwards, the blue dye was extracted from some lichens, especially in the Netherlands. | 0 | Chromatography + Titration + pH indicators |