Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space; when n = 3, the set of all such locations is called three-dimensional Euclidean space. It is represented by the symbol ℝ3; this serves as a three-parameter model of the physical universe. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions, any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space. Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height and length. In mathematics, analytic geometry describes every point in three-dimensional space by means of three coordinates.
Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are labeled x, y, z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. See Euclidean space. Below are images of the above-mentioned systems. Two distinct points always determine a line. Three distinct points determine a unique plane. Four distinct points can either coplanar or determine the entire space. Two distinct lines can either be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
Two distinct planes are parallel. Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane. A hyperplane is a subspace of one dimension less than the dimension of the full space; the hyperplanes of a three-dimensional space are the two-dimensional subspaces. In terms of cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations, each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, so, are coplanar. A sphere in 3-space consists of the set of all points in 3-space at a fixed distance r from a central point P.
The solid enclosed by the sphere is called a ball. The volume of the ball is given by V = 4 3 π r 3. Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space ℝ4. If a point has coordinates, P x2 + y2 + z2 + w2 = 1 characterizes those points on the unit 3-sphere centered at the origin. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution; the plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane, perpendicular to the axis, is a circle. Simple examples occur. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex the point of intersection. However, if the generatrix and axis are parallel, the surface of revolution is a circular cylinder.
In analogy with the conic sections, the set of points whose cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0, where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero is called a quadric surface. There are six types of non-degenerate quadric surfaces: Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Elliptic paraboloid Hyperbolic paraboloidThe degenerate quadric surfaces are the empty set, a single point, a single li
Polarization is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, transverse sound waves in solids. In some types of transverse waves, the wave displacement is limited to a single direction, so these do not exhibit polarization. An electromagnetic wave such as light consists of a coupled oscillating electric field and magnetic field which are always perpendicular.
In linear polarization, the fields oscillate in a single direction. In circular or elliptical polarization, the fields rotate at a constant rate in a plane as the wave travels; the rotation can have two possible directions. Light or other electromagnetic radiation from many sources, such as the sun and incandescent lamps, consists of short wave trains with an equal mixture of polarizations. Polarized light can be produced by passing unpolarized light through a polarizer, which allows waves of only one polarization to pass through; the most common optical materials are isotropic and do not affect the polarization of light passing through them. Some of these are used to make polarizing filters. Light is partially polarized when it reflects from a surface. According to quantum mechanics, electromagnetic waves can be viewed as streams of particles called photons; when viewed in this way, the polarization of an electromagnetic wave is determined by a quantum mechanical property of photons called their spin.
A photon has one of two possible spins: it can either spin in a right hand sense or a left hand sense about its direction of travel. Circularly polarized electromagnetic waves are composed of photons with only one type of spin, either right- or left-hand. Linearly polarized waves consist of photons that are in a superposition of right and left circularly polarized states, with equal amplitude and phases synchronized to give oscillation in a plane. Polarization is an important parameter in areas of science dealing with transverse waves, such as optics, seismology and microwaves. Impacted are technologies such as lasers and optical fiber telecommunications, radar. Most sources of light are classified as incoherent and unpolarized because they consist of a random mixture of waves having different spatial characteristics, frequencies and polarization states. However, for understanding electromagnetic waves and polarization in particular, it is easiest to just consider coherent plane waves. Characterizing an optical system in relation to a plane wave with those given parameters can be used to predict its response to a more general case, since a wave with any specified spatial structure can be decomposed into a combination of plane waves.
And incoherent states can be modeled stochastically as a weighted combination of such uncorrelated waves with some distribution of frequencies and polarizations. Electromagnetic waves, traveling in free space or another homogeneous isotropic non-attenuating medium, are properly described as transverse waves, meaning that a plane wave's electric field vector E and magnetic field H are in directions perpendicular to the direction of wave propagation. By convention, the "polarization" direction of an electromagnetic wave is given by its electric field vector. Considering a monochromatic plane wave of optical frequency f, let us take the direction of propagation as the z axis. Being a transverse wave the E and H fields must contain components only in the x and y directions whereas Ez = Hz = 0. Using complex notation, the instantaneous physical electric and magnetic fields are given by the real parts of the complex quantities occurring in the following equations; as a function of time t and spatial position z these complex fields can be written as: E → =
A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimension, a dihedral angle represents the angle between two hyperplanes; when the two intersecting planes are described in terms of Cartesian coordinates by the two equations a 1 x + b 1 y + c 1 z + d 1 = 0 a 2 x + b 2 y + c 2 z + d 2 = 0 the dihedral angle, φ between them is given by: cos φ = | a 1 a 2 + b 1 b 2 + c 1 c 2 | a 1 2 + b 1 2 + c 1 2 a 2 2 + b 2 2 + c 2 2. An alternative method is to calculate the angle between the vectors, nA and nB, which are normal to the planes. Cos φ = | n A ⋅ n B | | n A | | n B | where nA · nB is the dot product of the vectors and |nA| |nB| is the product of their lengths. Any plane can be described by two non-collinear vectors lying in that plane. Thus, a dihedral angle can be defined by three vectors, b1, b2 and b3, forming two pairs of non-collinear vectors.
Φ = atan2 . In chemistry, a torsion angle is defined as a particular example of a dihedral angle, describing the geometric relation of two parts of a molecule joined by a chemical bond; every set of three not-colinear atoms of a molecule defines a plane. When two such planes intersect, the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation. Stereochemical arrangements corresponding to angles between 0° and ±90° are called syn, those corresponding to angles between ±90° and 180° anti. Arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal and those between 0° and ±30° or ±150° and 180° are called periplanar; the two types of terms can be combined so as to define four ranges of angle. The synperiplanar conformation is known as the syn- or cis-conformation. For example, with n-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms; the syn-conformation shown above, with a dihedral angle of 60° is less stable than the anti-conformation with a dihedral angle of 180°.
For macromolecular usage the symbols T, C, G+, G−, A+ and A− are recommended. A Ramachandran plot developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, V. Sasisekharan, is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein st
In chemistry, a molecule experiences strain when its chemical structure undergoes some stress which raises its internal energy in comparison to a strain-free reference compound. The internal energy of a molecule consists of all the energy stored within it. A strained molecule has an additional amount of internal energy which an unstrained molecule does not; this extra internal energy, or strain energy, can be likened to a compressed spring. Much like a compressed spring must be held in place to prevent release of its potential energy, a molecule can be held in an energetically unfavorable conformation by the bonds within that molecule. Without the bonds holding the conformation in place, the strain energy would be released; the equilibrium of two molecular conformations is determined by the difference in Gibbs free energy of the two conformations. From this energy difference, the equilibrium constant for the two conformations can be determined. K e q = exp If there is a decrease in Gibbs free energy from one state to another, this transformation is spontaneous and the lower energy state is more stable.
A strained, higher energy molecular conformation will spontaneously convert to the lower energy molecular conformation. Enthalpy and entropy are related to Gibbs free energy through the equation: Δ G ∘ = Δ H ∘ − T Δ S ∘. Enthalpy is the more important thermodynamic function for determining a more stable molecular conformation. While there are different types of strain, the strain energy associated with all of them is due to the weakening of bonds within the molecule. Since enthalpy is more important, entropy can be ignored; this isn't always the case. For example, n-butane has two possible conformations and gauche; the anti conformation is more stable by 0.9 kcal mol−1. We would expect that butane is 82% anti and 18% gauche at room temperature. However, there are only one anti conformation. Therefore, entropy makes a contribution of 0.4 kcal in favor of the gauche conformation. We find that the actual conformational distribution of butane is 70% anti and 30% gauche at room temperature; the standard heat of formation of a compound is described as the enthalpy change when the compound is formed from its separated elements.
When the heat of formation for a compound is different from either a prediction or a reference compound, this difference can be attributed to strain. For example, ΔfH° for cyclohexane is -29.9 kcal mol−1 while ΔfH° for methylcyclopentane is -25.5 kcal mol−1. Despite having the same atoms and number of bonds, methylcyclopentane is higher in energy than cyclohexane; this difference in energy can be attributed to the ring strain of a five-membered ring, absent in cyclohexane. Experimentally, strain energy is determined using heats of combustion, an easy experiment to perform. Determining the strain energy within a molecule requires knowledge of the expected internal energy without the strain. There are two ways. First, one could compare to a similar compound that lacks strain, such as in the previous methylcyclohexane example, it can be difficult to obtain a suitable compound. An alternative is to use Benson group increment theory; as long as suitable group increments are available for the atoms within a compound, a prediction of ΔfH° can be made.
If the experimental ΔfH° differs from the predicted ΔfH°, this difference in energy can be attributed to strain energy. Van der Waals strain, or steric strain, occurs when atoms are forced to get closer than their Van der Waals radii allow. Van der Waals strain is considered a form of strain where the interacting atoms are at least four bonds away from each other; the amount on steric strain in similar molecules is dependent on the size of the interacting groups. The effects of steric strain in the reaction of trialkylamines and trimethylboron were studied by Nobel laureate Herbert C. Brown et al, they found that as the size of the alkyl groups on the amine were increased, the equilibrium constant decreased as well. The shift in equilibrium was attributed to steric strain between the alkyl groups of the amine and the methyl groups on boron. There are situations where identical conformations are not equal in strain energy. Syn-pentane strain is an example of this situation. There are two different ways to put both of the bonds the central in n-pentane into a gauche conformation, one of, 3 kcal mol−1 higher in energy than the other.
When the two methyl-substituted bonds are rotated from anti to gauche in opposite directions, the molecule assumes a cyclopentane-like conformation where the two terminal methyl groups are brought into proximity. If the bonds are rotated in the same direction, this doesn't occur; the steric strain between the two terminal methyl groups accounts for the difference in energy between the two similar, yet different conformations. Allylic strain, or A1,3 strain is associated to syn-pentane strain. An example of allylic strain can be see
Chemistry is the scientific discipline involved with elements and compounds composed of atoms and ions: their composition, properties and the changes they undergo during a reaction with other substances. In the scope of its subject, chemistry occupies an intermediate position between physics and biology, it is sometimes called the central science because it provides a foundation for understanding both basic and applied scientific disciplines at a fundamental level. For example, chemistry explains aspects of plant chemistry, the formation of igneous rocks, how atmospheric ozone is formed and how environmental pollutants are degraded, the properties of the soil on the moon, how medications work, how to collect DNA evidence at a crime scene. Chemistry addresses topics such as how atoms and molecules interact via chemical bonds to form new chemical compounds. There are four types of chemical bonds: covalent bonds, in which compounds share one or more electron; the word chemistry comes from alchemy, which referred to an earlier set of practices that encompassed elements of chemistry, philosophy, astronomy and medicine.
It is seen as linked to the quest to turn lead or another common starting material into gold, though in ancient times the study encompassed many of the questions of modern chemistry being defined as the study of the composition of waters, growth, disembodying, drawing the spirits from bodies and bonding the spirits within bodies by the early 4th century Greek-Egyptian alchemist Zosimos. An alchemist was called a'chemist' in popular speech, the suffix "-ry" was added to this to describe the art of the chemist as "chemistry"; the modern word alchemy in turn is derived from the Arabic word al-kīmīā. In origin, the term is borrowed from the Greek χημία or χημεία; this may have Egyptian origins since al-kīmīā is derived from the Greek χημία, in turn derived from the word Kemet, the ancient name of Egypt in the Egyptian language. Alternately, al-kīmīā may derive from χημεία, meaning "cast together"; the current model of atomic structure is the quantum mechanical model. Traditional chemistry starts with the study of elementary particles, molecules, metals and other aggregates of matter.
This matter can be studied in isolation or in combination. The interactions and transformations that are studied in chemistry are the result of interactions between atoms, leading to rearrangements of the chemical bonds which hold atoms together; such behaviors are studied in a chemistry laboratory. The chemistry laboratory stereotypically uses various forms of laboratory glassware; however glassware is not central to chemistry, a great deal of experimental chemistry is done without it. A chemical reaction is a transformation of some substances into one or more different substances; the basis of such a chemical transformation is the rearrangement of electrons in the chemical bonds between atoms. It can be symbolically depicted through a chemical equation, which involves atoms as subjects; the number of atoms on the left and the right in the equation for a chemical transformation is equal. The type of chemical reactions a substance may undergo and the energy changes that may accompany it are constrained by certain basic rules, known as chemical laws.
Energy and entropy considerations are invariably important in all chemical studies. Chemical substances are classified in terms of their structure, phase, as well as their chemical compositions, they can be analyzed using the tools of e.g. spectroscopy and chromatography. Scientists engaged in chemical research are known as chemists. Most chemists specialize in one or more sub-disciplines. Several concepts are essential for the study of chemistry; the particles that make up matter have rest mass as well – not all particles have rest mass, such as the photon. Matter can be a mixture of substances; the atom is the basic unit of chemistry. It consists of a dense core called the atomic nucleus surrounded by a space occupied by an electron cloud; the nucleus is made up of positively charged protons and uncharged neutrons, while the electron cloud consists of negatively charged electrons which orbit the nucleus. In a neutral atom, the negatively charged electrons balance out the positive charge of the protons.
The nucleus is dense. The atom is the smallest entity that can be envisaged to retain the chemical properties of the element, such as electronegativity, ionization potential, preferred oxidation state, coordination number, preferred types of bonds to form. A chemical element is a pure substance, composed of a single type of atom, characterized by its particular number of protons in the nuclei of its atoms, known as the atomic number and represented by the symbol Z; the mass number is the sum of the number of neutrons in a nucleus. Although all the nuclei of all atoms belonging to one element will have the same
Biochemistry, sometimes called biological chemistry, is the study of chemical processes within and relating to living organisms. Biochemical processes give rise to the complexity of life. A sub-discipline of both biology and chemistry, biochemistry can be divided in three fields. Over the last decades of the 20th century, biochemistry has through these three disciplines become successful at explaining living processes. All areas of the life sciences are being uncovered and developed by biochemical methodology and research. Biochemistry focuses on understanding how biological molecules give rise to the processes that occur within living cells and between cells, which in turn relates to the study and understanding of tissues and organism structure and function. Biochemistry is related to molecular biology, the study of the molecular mechanisms by which genetic information encoded in DNA is able to result in the processes of life. Much of biochemistry deals with the structures and interactions of biological macromolecules, such as proteins, nucleic acids and lipids, which provide the structure of cells and perform many of the functions associated with life.
The chemistry of the cell depends on the reactions of smaller molecules and ions. These can be inorganic, for example water and metal ions, or organic, for example the amino acids, which are used to synthesize proteins; the mechanisms by which cells harness energy from their environment via chemical reactions are known as metabolism. The findings of biochemistry are applied in medicine and agriculture. In medicine, biochemists investigate the cures of diseases. In nutrition, they study how to maintain health wellness and study the effects of nutritional deficiencies. In agriculture, biochemists investigate soil and fertilizers, try to discover ways to improve crop cultivation, crop storage and pest control. At its broadest definition, biochemistry can be seen as a study of the components and composition of living things and how they come together to become life, in this sense the history of biochemistry may therefore go back as far as the ancient Greeks. However, biochemistry as a specific scientific discipline has its beginning sometime in the 19th century, or a little earlier, depending on which aspect of biochemistry is being focused on.
Some argued that the beginning of biochemistry may have been the discovery of the first enzyme, diastase, in 1833 by Anselme Payen, while others considered Eduard Buchner's first demonstration of a complex biochemical process alcoholic fermentation in cell-free extracts in 1897 to be the birth of biochemistry. Some might point as its beginning to the influential 1842 work by Justus von Liebig, Animal chemistry, or, Organic chemistry in its applications to physiology and pathology, which presented a chemical theory of metabolism, or earlier to the 18th century studies on fermentation and respiration by Antoine Lavoisier. Many other pioneers in the field who helped to uncover the layers of complexity of biochemistry have been proclaimed founders of modern biochemistry, for example Emil Fischer for his work on the chemistry of proteins, F. Gowland Hopkins on enzymes and the dynamic nature of biochemistry; the term "biochemistry" itself is derived from a combination of chemistry. In 1877, Felix Hoppe-Seyler used the term as a synonym for physiological chemistry in the foreword to the first issue of Zeitschrift für Physiologische Chemie where he argued for the setting up of institutes dedicated to this field of study.
The German chemist Carl Neuberg however is cited to have coined the word in 1903, while some credited it to Franz Hofmeister. It was once believed that life and its materials had some essential property or substance distinct from any found in non-living matter, it was thought that only living beings could produce the molecules of life. In 1828, Friedrich Wöhler published a paper on the synthesis of urea, proving that organic compounds can be created artificially. Since biochemistry has advanced since the mid-20th century, with the development of new techniques such as chromatography, X-ray diffraction, dual polarisation interferometry, NMR spectroscopy, radioisotopic labeling, electron microscopy, molecular dynamics simulations; these techniques allowed for the discovery and detailed analysis of many molecules and metabolic pathways of the cell, such as glycolysis and the Krebs cycle, led to an understanding of biochemistry on a molecular level. Philip Randle is well known for his discovery in diabetes research is the glucose-fatty acid cycle in 1963.
He confirmed. High fat oxidation was responsible for the insulin resistance. Another significant historic event in biochemistry is the discovery of the gene, its role in the transfer of information in the cell; this part of biochemistry is called molecular biology. In the 1950s, James D. Watson, Francis Crick, Rosalind Franklin, Maurice Wilkins were instrumental in solving DNA structure and suggesting its relationship with genetic transfer of information. In 1958, George Beadle and Edward Tatum received the Nobel Prize for work in fungi showing that one gene produces one enzyme. In 1988, Colin Pitchfork was the first person convicted of murder with DNA evidence, which led to the growth of forensic science. More Andrew Z. Fire and Craig C. Mello received the 2006 Nobel Prize for discovering the role of RNA interference, in the silencing of gene expression. Around two dozen of the 92
Solid-state chemistry sometimes referred as materials chemistry, is the study of the synthesis and properties of solid phase materials but not exclusively of, non-molecular solids. It therefore has a strong overlap with solid-state physics, crystallography, metallurgy, materials science and electronics with a focus on the synthesis of novel materials and their characterisation. Solids can be classified as crystalline or amorphous on basis of the nature of order present in the arrangement of their constituent particles; because of its direct relevance to products of commerce, solid state inorganic chemistry has been driven by technology. Progress in the field has been fueled by the demands of industry, sometimes in collaboration with academia. Applications discovered in the 20th century include zeolite and platinum-based catalysts for petroleum processing in the 1950s, high-purity silicon as a core component of microelectronic devices in the 1960s, “high temperature” superconductivity in the 1980s.
The invention of X-ray crystallography in the early 1900s by William Lawrence Bragg was an enabling innovation. Our understanding of how reactions proceed at the atomic level in the solid state was advanced by Carl Wagner's work on oxidation rate theory, counter diffusion of ions, defect chemistry; because of his contributions, he has sometimes been referred to as the father of solid state chemistry. Given the diversity of solid state compounds, an diverse array of methods are used for their preparation. For organic materials, such as charge transfer salts, the methods operates near room temperature and are similar to the techniques of organic synthesis. Redox reactions are sometimes conducted by electrocrystallisation, as illustrated by the preparation of the Bechgaard salts from tetrathiafulvalene. For thermally robust materials, high temperature methods are employed. For example, bulk solids are prepared using tube furnaces, which allow reactions to be conducted up to ca. 1100 °C. Special equipment e.g. ovens consisting of a tantalum tube through which an electric current is passed can be used for higher temperatures up to 2000 °C.
Such high temperatures are at times required to induce diffusion of the reactants. One method employed is to melt the reactants together and later anneal the solidified melt. If volatile reactants are involved the reactants are put in an ampoule, evacuated -ofnt mixture cold e.g. by keeping the bottom of the ampoule in liquid nitrogen- and sealed. The sealed ampoule is put in an oven and given a certain heat treatment.. It is possible to use solvents to prepare solids by evaporation. At times the solvent is used hydrothermal, under pressure at temperatures higher than the normal boiling point. A variation on this theme is the use of flux methods, where a salt of low melting point is added to the mixture to act as a high temperature solvent in which the desired reaction can take place; this can be useful Many solids react vigorously with reactive gas species like chlorine, oxygen etc. Others form adducts with e.g. CO or ethylene; such reactions are conducted in a tube, open ended on both sides and through which the gas is passed.
A variation of this is to let the reaction take place inside a measuring device such as a TGA. In that case stoichiometric information can be obtained during the reaction, which helps identify the products. A special case of a gas reaction is a chemical transport reaction; these are carried out in a sealed ampoule to which a small amount of a transport agent, e.g. iodine is added. The ampoule is placed in a zone oven; this is two tube ovens attached to each other which allows a temperature gradient to be imposed. Such a method can be used to obtain the product in the form of single crystals suitable for structure determination by X-ray diffraction. Chemical vapour deposition is a high temperature method, employed for the preparation of coatings and semiconductors from molecular precursors. Synthetic methodology and characterization go hand in hand in the sense that not one but a series of reaction mixtures are prepared and subjected to heat treatment; the stoichiometry is varied in a systematic way to find which stoichiometries will lead to new solid compounds or to solid solutions between known ones.
A prime method to characterize the reaction products is powder diffraction, because many solid state reactions will produce polycristalline ingots or powders. Powder diffraction will facilitate the identification of known phases in the mixture. If a pattern is found, not known in the diffraction data libraries an attempt can be made to index the pattern, i.e. to identify the symmetry and the size of the unit cell. Once the unit cell of a new phase is known, the next step is to establish the stoichiometry of the phase; this can be done in a number of ways. Sometimes the composition of the original mixture will give a clue, if one finds only one product -a single powder pattern- or if one was trying to make a phase of a certain composition by analogy to known materials but this is rare. Considerable effort in refining the synthetic methodology is required to obtain a pure sample of the new material. If it is possible to separate the product from the rest of the reaction mixture elemental analysis can be used.
Another way involves the generation of characteristic X-rays in the electron beam. X-ray diffraction is used due to its imaging capabilities and speed of data generation; the latter requires revisiting and ref