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Anisotropic Displacement Parameter |
A description of the thermal vibration or other state of motion, of an atom in a crystal. The uncertainty in the atomic position is represented by a three-dimensional shape, the surface of which defines a fixed probability of the atom being located within. For harmonic motions of the atom, this is an ellipsoid - a "thermal ellipsoid". The ellipsoid is represented by a symmetric tensor, with six unique components (Uij), giving both the magnitude, and the direction of the atom's displacement from its ideal position - hence the term "atomic displacement parameters". Also referred to as the "anisotropic temperature factor". |
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Asymmetric Unit |
The term "asymmetric unit" refers to the smallest group of atoms required to uniquely define a crystal structure. To define a crystal, one could list all the atoms inside one unit cell. However, if the crystal has some symmetry then some of these atoms will be related to other atoms by the operation of symmetry elements. So, a more concise way of defining a crystal structure is to specify the crystal symmetry (i.e., its spacegroup), and then give the smallest subset of the unit cell (i.e., the asymmetric unit) which, when combined with the symmetry "generates" the remainder of the unit cell. |
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Atomic Displacement Parameter | See Thermal Ellipsoid. | ||||||||||||||||||||||||||||||||
Atomic Scattering Factor |
X-rays are scattered by the electrons of an atom. Because the diameter of electron clouds is similar to the wavelength of x-rays, the amount of scattering varies with angle. The amplitude of x-ray radiation scattered by a single atom is defined by the atomic scattering factor (ASF). Its angular dependence is summzrized in a 9-coefficient polynomial equation. Values for the ASF coefficients may be found in the International Tables for Crystallography, or in: Doyle & Turner (1968) Acta Crystallographica A24:390-397. |
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Bragg Equation |
The most important equation in the history of diffraction was derived by W L Bragg in 1912 and relates the angle at which radiation is scattered by a set of lattice planes to the inter-planar separation (d-spacing) and the wavelength of the radiation: λ = 2 d sin θ where λ is the wavelength, d is the d-spacing of the lattice planes, and θ (the Bragg Angle) is the inclination of the incident radiation beam to the surface of the lattice planes. |
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Bravais Lattices |
The 14 unique three-dimensional lattice types - as derived by the German crystallographer, Bravais, in the mid-nineteenth century. He showed that, when considering all combinations of lattice types and crystal systems, there was a total of 14 unique lattice, which have become known as the "Bravais Lattices". |
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Cell Parameters |
See Lattice Parameters. |
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Crystal |
A crystal is a solid material which has a regular internal arrangement of atoms, which sets limits on its range of chemical compositions, and gives it a characteristic crystal shape. The internal arrangement of atoms can be resolved into an endlessly-repeated unit cell, which contains one or more atoms. The relative arrangment of atoms within the unit cell defines the structure's symmetry, as summarized by the crystallographic spacegroup. |
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Crystal System |
A crystal system relates to the shape and symmetry of the unit cell, and in particular, to the relative lengths and the angles between the cell edges. For example, the cubic system is defined by a cube-shaped unit cell, where the cell edges a, b and c are mutually perpendicular, and have the same lengths. On the other hand, the triclinic system is characterized by a "distorted" unit cell, in which the cell edge lengths and their interaxial angles may all be different. The 7 unique crystal systems are summarized below:-
(*) Note that trigonal crystals can be described in terms of an hexagonal unit cell, although their maximum symmetry is 3-fold, not 6-fold. |
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Debye-Waller Factor |
The Debye-Waller factor describes the amount by which scattered intensities are reduced by thermal vibrations of atoms. It has the form: exp( —Biso sin2 θ / λ2 ) where θ is the scattering angle, λ is the wavelength, and Biso is the atom's temperature factor (related to the atom's isotropic displacement parameter by the relation: Biso = 8 π2 Uiso.) |
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Delta Function |
A delta function is an infinitely-narrow peak, with a finite intensity. For CrystalDiffract, lines are drawn at the positions of each calculated peak, with heights corresponding to the calculated, integrated intensity. The delta function may be useful because it:
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Diffraction |
Diffraction is the process by which waves interact with objects and are disturbed, or bent, from their previous course (e.g., the movement of water ripples around the end of a pier). In X-ray crystallography, the diffraction of certain electromagnetic waves (X-rays) from atoms in a crystal is studied. In neutron diffraction, a high-energy beam of neutrons (nuclear particles with no charge) has wave-like properties and can also be diffracted from atoms in a crystal lattice. |
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D-Spacing |
The term "d-spacing" refers to the inter-planar separation for a set of lattice planes: that is, the distance between successive planes, measured along the plane normal (at 90 degrees to each lattice plane). Planes of atoms in a crystal act as miniature diffraction gratings, and their d-spacings control the angles at which radiation is scattered, as summarized in the famous Bragg Equation of 1912. |
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Energy-Dispersive |
A relatively-new type of diffraction involves using "white" radiation that has a spread of wavelengths. Whereas a a fixed-wavelength experiment requires one to mechanically scan a detector through a range of θ/2θ angles (see the Bragg Equation), using "white" radiation allows the use of a stationary detector, with diffraction from a range of d-spacings being recorded at the same θ angle. The different scattered beams have different wavelengths, and hence energies - hence this form of diffraction is referred to as "energy-dispersive". CrystalDiffract allows you to specify the 2θ value for the diffraction experiment, using the Diffract > Energy command. You can also adjust this interactively, using the Inspector. |
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Eta |
The Eta (η) parameter describes the shape of a Pseudo-Voigt peak profile. It varies between 0 (a pure Gaussian peak) and 1 (a pure Lorentzian peak). |
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Form Factor | |||||||||||||||||||||||||||||||||
Fractional Coordinates |
For crystal structures it is conventional to define atomic coordinates relative to the unit cell dimensions; these relative coordinates are expressed as fractions of the unit cell dimensions along the crystallographic x, y, and z axes. For example, if an atom in a cubic crystal has orthogonal coordinates {x = 2 Å, y = 4 Å, z = 6 Å} and the unit cell parameter a (= b = c) is 10 Å, then its fractional coordinates are {0.2, 0.4, 0.6}. |
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Gaussian |
The Gaussian function is a mathematical function, characterized by a bell-shaped peak ("bell curve"), without the wide tail of a Lorentzian function. |
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General Equivalent Position |
Crystallographic symmetry includes lattice translations (defined by the lattice type), plus additional symmetry operations associated with each lattice point. These symmetry operations, which may include mirror or rotational symmetry, can be summarized by listing the relative coordinates to which a single atom is mapped. For example, consider a crystal whose unit cell contains just a single symmetry element: a mirror plane parallel to the x and z axes, and passing through the y axis at the origin. The mirror plane will reflect (or map) a point at {+x, +y, +z} to the position at {+x, -y, +z}. In other words, the positions {+x, +y, +z} and {+x, -y, +z} are symmetrically-equivalent - and are referred to as "general equivalent positions" for this crystal. |
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Graphs and Films |
CrystalDiffract can display diffraction patterns in either "graph" mode, or in "film" mode. CrystalDiffract graphs represent how the intensity of radiation diffracted by a crystal varies according to scattering angle (2θ), time-of-flight, or some other parameter, plotted along the graph's horizontal (x) axis. The intensity is plotted along the graph's vertical (y) axis. The "taller" the peaks, the more intense the diffraction. In film mode, CrystalDiffract simulates the appearance of a photographic film (negative) which as been wrapped around a crystal exposed to radiation. Reflexions appear as dark lines on the grey film background. The darker the line, the more intense the reflexion. |
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Indexing Peaks |
"Indexing peaks" is the process of assigning Miller Indices to diffraction peaks, to signify which sets of planes caused which reflection. |
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Intensity |
Crystals are regular, ordered structures, and contain billions of atoms arranged in regularly-spaced lattice planes. These planes act as diffraction gratings for radiation whose wavelength is similar to the lattice plane spacing (e.g., x-rays and neutrons can be diffracted by crystals). The strength of diffraction from a particular set of lattice planes is termed the intensity of reflection. This depends on the type and arrangement of atoms parallel to the lattice planes; atoms lying between the lattice planes are "out of phase" and reduce the intensity of the scattering. Knowing the orientation of a lattice plane (which is determined by its Miller Indices), and details of the crystal structure (its lattice parameters, symmetry, and asymmetric unit), CrystalDiffract can calculate the intensity for any reflexion, based on a fixed volume. (Note that in a real experiment, the absolute intensity will depend on a multitude of factors, including the radiation intensity, sample volume, detector efficiency, etc.) |
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Isotropic Displacement Parameter |
This is a representation of the thermal (or other) motion of an atom, assuming equal motions in all directions. The radius of the sphere is the atom's mean-square displacement, < u2 >, written Uiso (in units of Å2). N.B., the isotropic displacement parameter, Uiso is related to the atom's isotropic temperature factor, Biso by a factor of 8 π2: Biso = 8 π2 Uiso |
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Lattice |
A lattice is defined as a regular arrangement of points throughout space. This is a mathematical concept, used to describe the regular repetition, throughout 3-d space of individual atoms, or groups of atoms (i.e., the asymmetric unit). The basic building brick of a crystal, the unit cell, has lattice points at each of its eight corners. It may also have lattice points at the centres of certain faces, or at the centre of the unit cell. This arrangement of lattice points within the unit cell defines the lattice type of a crystal: P (Primitive); I (Body-centred, from the German Innenzentrierte; "inner-centred"); A-, B- and C-face centred; F (all-face-centred) and R (Rhombohedral). |
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Lattice Parameters |
Six numbers which uniquely define the size and shape of the unit cell. These are the lengths of the unit cell edges, a, b, c; and the inter-axial angles α, β, and γ. |
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Lattice Plane |
A lattice plane is a plane of lattice points in a three-dimensional structure. The orientation of a lattice plane can be given in terms of its intercepts along the unit cell axes xyz. These are summarized using Miller Indices, (hkl). |
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Lattice Vector |
A lattice vector is defined as a vector between two lattice points. The orientation of the vector can be expressed in terms of its coefficients U, V, and W, parallel to the unit cell vectors a, b, and c, respectively, giving the vector equation: Ua + Vb + Wc which is generally written in a shorthand form as: [UVW] Note that the unit cell vectors are not necessarily mutually orthogonal, nor may they be of equal length. So, for example, in an orthorhombic crystal (a ≠ b ≠ c), the lattice vector [100] probably has a different length from the vector [010]. |
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Lorentzian |
The Lorenztian function is a mathematical function, characterized by trumpet-like peaks, with very wide tails. The Lorentzian function generally provides a good description of the shape of neutron diffraction peaks. |
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Lorentz factor |
A geometric factor that affects the intensity of a reflexion. The Lorentz factor takes into account the time taken for a reflexion to be in a scattering position. Mathematically, this is represented by the time taken for a reciprocal lattice point to pass through the surface of the reflecting sphere (the "Ewald Sphere"). This depends on the scattering angle, as well as the experimental geometry. |
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Miller Indices |
Miller Indices are sets of three integers, designated h, k, l, which uniquely define the orientation of a lattice plane, relative to a crystal's unit cell. A lattice plane's orientation can be defined in terms of its intersections with the three crystallographic axes, x, y, and z. If a unit cell has edge lengths a, b, and c, and the first lattice plane from the origin cuts these axes at a/h, b/k, and c/l, then the Miller Indices of that plane are: (h k l). |
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Monochromatic |
Monochromatic radiation is radiation characterized by a single x-ray or neutron wavelength (e.g., "1.54 Å"). Many synchrotron x-ray sources and neutron diffractometers produce monochromatic radiation. In contrast, traditional laboratory x-ray tubes usually produce dual-wavelength radiation. The two wavelengths correspond to Kα1 and Kα2 x-ray emission peaks from the target material. |
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Multiplicity |
"Multiplicity" refers to the number of X-ray or neutron reflexions that contribute to a single peak observed in a powder pattern. Multiple reflexions with the same d-spacing become superimposed in a one-dimensional x-ray powder pattern, giving rise to a single peak, with higher intensity. The total number of reflexions contributing to that peak is the peak's "multiplicity". |
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Neutron Scattering Length |
Neutrons are scattered by atomic nuclei, which are much smaller than neutron wavelengths - so the scattering is essentially independent of angle. The amplitude scattered by an atomic nucleus is represented by the neutron scattering length. |
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Percent Strain |
A measurement of the amount of strain in a powdered crystalline sample. The percent strain is the standard deviation for the variation of cell parameters in the sample (in an ideal crystal there would be one unique cell parameter, whereas in a strained crystal there might be a normal distribution of cell parameter values, characterized by a standard deviation, ranging from zero for the ideal crystal to a few percent for a very-highly strained crystal). |
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Polarization Factor |
A factor that takes into account the reduction in intensity caused by the polarization of (X-ray) radiation when scattered by electrons. |
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Preferred Orientation |
A non-random distribution of crystallites in the sample (e.g., the tendency of platy crystals to lie flat parallel to the specimen holder, or for needles to clump together parallel to their axes). Preferred orientation causes some reflexions to be scattered more strongly, whilst other reflexions will appear weaker - if at all. |
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Primitive Lattice |
The most basic type of lattice, where the unit cell has lattice points only at its corners. Since each corner is shared between eight unit cells, this gives a total of 8 x 1/8 = 1 lattice point per unit cell. |
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Pseudo-Voigt |
The Pseudo-Voigt function is a mathematical function used to describe the shape of diffraction peaks. It is defined as the weighted sum of a Lorentzian and a Gaussian peak. The Lorentzian fraction is defined by the Eta parameter. |
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Reflexion |
A peak of intensity, caused by the scattering of a beam of radiation, by planes of atoms in a crystal. |
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Rhombohedral |
This is a special kind of Primitive lattice, where the body diagonal has three-fold symmetry and the inter-axial angles are all non-orthogonal. |
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Site Occupancy |
The term "site occupancy" refers to the chemical content of a crystallographic site. In an ideal structure, a site in the crystal might contain one specific chemical element per site, e.g., a copper (Cu) atom on site 1 and a zinc (Zn) atom on site 2. Sometimes, however, crystals show disordered structures and there might be a statistical probability of different atoms occupying the same sites. For example 50% Cu and 50% Zn on sites 1 and 2. The site occupancy for both sites would then be written Cu0.50 Zn0.50. |
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Spacegroup |
A spacegroup is a three-dimensional arrangement of symmetry elements, positioned within a crystal's unit cell, and relating atoms in the asymmetric unit (or "motif") to the rest of the atoms in the unit cell. A spacegroup can be represented by a spacegroup symbol: a descriptive tag which may or may not give further information about the precise arrangement of symmetry elements. Crystallographers commonly use the "International Notation" of spacegroup symbols, examples of which include "P 1", "P 2/c", "I a -3 d", etc. The first character of such a spacegroup symbol (e.g., "P" or "I") is the lattice type; the remaining characters relate to symmetry operators ("2" is a 2-fold rotation axis; "c" is a c-glide plane perpendicular to the 2-fold axis; etc.) |
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Structure Factor |
A Structure Factor, F(hkl) describes the amplitude and phase of radiation scattered by one unit cell of material for the set of lattice planes (hkl). The Structure Factor is a complex number, with real (A) and imaginary (B) parts:- F(hkl) = A(hkl) + i B(hkl) The phase angle, φ, is given by the ratio of the imaginary to real parts, that is:- tan φ = B / A The Intensity of the scattered radiation for the reflection (hkl) is proportional to the square of the amplitude, | F |2. |
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Temperature Factors |
See Thermal Ellipsoid. |
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Thermal Ellipsoid |
Thermal ellipsoids (also referred to as "atomic displacement parameters", or "temperature factors") are used to describe the relative magnitude, and orientation, of atomic motions (or positional disorder) in a crystal. In most structures the atoms are not at rest, but vibrate about their mean positions. A statistical analysis of these motions, assuming harmonic vibrations, confines an atom centre to a probability surface whose three-dimensional representation is, in its most general case, an ellipsoid (described by a anisotropic displacement parameters), sometimes approximated by a sphere (described by an isotropic displacement parameter). |
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Time-of-Flight |
A type of neutron diffraction experiment in which scattered intensities are recorded as a function of the time taken for neutrons to reach a fixed detector. A neutron spallation source (such as ISIS at the Rutherford-Appleton Laboratory, near Oxford, England) creates pulses of neutrons with a range of energies. These travel at different speeds, depending on the energy of the neutrons, and are directed down a long "beam line" towards a powder sample. Diffraction is recorded by neutron detectors arranged around the sample, at a fixed 2θ angle. The number of pulses is recorded as a function of the time-of-flight of the neutrons (which is typically in the range of a few milliseconds to several hundred milliseconds). As for energy-dispersive diffraction, an extended diffraction pattern can be recorded at a fixed angle because the sample is subjected to neutrons of different energies, and hence wavelengths. |
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Unit Cell |
The basic building brick of a crystal structure. The unit cell is a parallelipiped with edges parallel to the three crystallographic axes, x, y, and z. The edges along the x, y, and z directions define the crystallographic basis vectors, a, b, and c, respectively. The inter-axial angles are, by convention, called α (= y^z), β (= x^z) and γ (= x^y). The axial dimensions and inter-axial angles uniquely define the three-dimensional periodic nature of the crystal's lattice, and are known as the lattice parameters. |