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91-406: Crystallography is a broad topic, and many of its subareas, such as X-ray crystallography , are themselves important scientific topics. Crystallography ranges from the fundamentals of crystal structure to the mathematics of crystal geometry , including those that are not periodic or quasicrystals . At the atomic scale it can involve the use of X-ray diffraction to produce experimental data that

182-443: A Faraday cup . Also, since LEED is a surface-sensitive method, it required well-ordered surface structures. Techniques for the preparation of clean metal surfaces first became available much later. Nonetheless, H. E. Farnsworth and coworkers at Brown University pioneered the use of LEED as a method for characterizing the absorption of gases onto clean metal surfaces and the associated regular adsorption phases, starting shortly after

273-441: A Wulff net or Lambert net . The pole to each face is plotted on the net. Each point is labelled with its Miller index . The final plot allows the symmetry of the crystal to be established. The discovery of X-rays and electrons in the last decade of the 19th century enabled the determination of crystal structures on the atomic scale, which brought about the modern era of crystallography. The first X-ray diffraction experiment

364-505: A Varian system. It soon became clear that the kinematic (single-scattering) theory, which had been successfully used to explain X-ray diffraction experiments, was inadequate for the quantitative interpretation of experimental data obtained from LEED. At this stage a detailed determination of surface structures, including adsorption sites, bond angles and bond lengths was not possible. A dynamical electron-diffraction theory, which took into account

455-424: A ccp arrangement of atoms is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the {111} planes of the fcc unit cell. There are four different orientations of the close-packed layers. One important characteristic of a crystalline structure is its atomic packing factor (APF). This is calculated by assuming that all the atoms are identical spheres, with

546-624: A certain degree of surface symmetries. However it will give no information about the atomic arrangement within a surface unit cell or the sites of adsorbed atoms. For instance, when the whole superstructure in Figure 7 is shifted such that the atoms adsorb in bridge sites instead of on-top sites the LEED pattern stays the same, although the individual spot intensities may somewhat differ. A more quantitative analysis of LEED experimental data can be achieved by analysis of so-called I–V curves, which are measurements of

637-599: A lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystallographic point group or crystal class

728-475: A particle with linear momentum p is given by h / p , where h is the Planck constant . The de Broglie hypothesis was confirmed experimentally at Bell Labs in 1927, when Clinton Davisson and Lester Germer fired low-energy electrons at a crystalline nickel target and observed that the angular dependence of the intensity of backscattered electrons showed diffraction patterns. These observations were consistent with

819-413: A radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres which can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows: Another important characteristic of a crystalline structure is its coordination number (CN). This is the number of nearest neighbours of a central atom in

910-493: A set of trial structures is created by varying the model parameters. The parameters are changed until an optimal agreement between theory and experiment is achieved. However, for each trial structure a full LEED calculation with multiple scattering corrections must be conducted. For systems with a large parameter space the need for computational time might become significant. This is the case for complex surfaces structures or when considering large molecules as adsorbates. Tensor LEED

1001-412: A simple hexagonal surface of a metal has been covered with a layer of graphene . Figure 7 shows a schematic of real and reciprocal space lattices for a simple (1×2) superstructure on a square lattice. For a commensurate superstructure the symmetry and the rotational alignment with respect to adsorbent surface can be determined from the LEED pattern. This is easiest shown by using a matrix notation, where

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1092-474: A single crystal, but are poly-crystalline in nature (they exist as an aggregate of small crystals with different orientations). As such, powder diffraction techniques, which take diffraction patterns of samples with a large number of crystals, play an important role in structural determination. Other physical properties are also linked to crystallography. For example, the minerals in clay form small, flat, platelike structures. Clay can be easily deformed because

1183-400: A vector normal to the plane. Considering only ( hkℓ ) planes intersecting one or more lattice points (the lattice planes ), the distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula The crystallographic directions are geometric lines linking nodes ( atoms , ions or molecules ) of a crystal. Likewise,

1274-505: Is "at infinity"). A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined. The Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in (1 2 3). In an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of

1365-781: Is a freely accessible repository for the structures of proteins and other biological macromolecules. Computer programs such as RasMol , Pymol or VMD can be used to visualize biological molecular structures. Neutron crystallography is often used to help refine structures obtained by X-ray methods or to solve a specific bond; the methods are often viewed as complementary, as X-rays are sensitive to electron positions and scatter most strongly off heavy atoms, while neutrons are sensitive to nucleus positions and scatter strongly even off many light isotopes, including hydrogen and deuterium. Electron diffraction has been used to determine some protein structures, most notably membrane proteins and viral capsids . The International Tables for Crystallography

1456-422: Is a technique for the determination of the surface structure of single-crystalline materials by bombardment with a collimated beam of low-energy electrons (30–200 eV) and observation of diffracted electrons as spots on a fluorescent screen. LEED may be used in one of two ways: An electron-diffraction experiment similar to modern LEED was the first to observe the wavelike properties of electrons, but LEED

1547-559: Is achieved. A quantitative measure for this agreement is the so-called reliability - or R-factor. A commonly used reliability factor is the one proposed by Pendry. It is expressed in terms of the logarithmic derivative of the intensity: The R-factor is then given by: where Y ( E ) = L − 1 / ( L − 2 + V o i 2 ) {\displaystyle Y(E)=L^{-1}/(L^{-2}+V_{oi}^{2})} and V o i {\displaystyle V_{oi}}

1638-526: Is an attempt to reduce the computational effort needed by avoiding full LEED calculations for each trial structure. The scheme is as follows: One first defines a reference surface structure for which the I–V spectrum is calculated. Next a trial structure is created by displacing some of the atoms. If the displacements are small the trial structure can be considered as a small perturbation of the reference structure and first-order perturbation theory can be used to determine

1729-404: Is an eight-book series that outlines the standard notations for formatting, describing and testing crystals. The series contains books that covers analysis methods and the mathematical procedures for determining organic structure through x-ray crystallography, electron diffraction, and neutron diffraction. The International tables are focused on procedures, techniques and descriptions and do not list

1820-402: Is considered a bad agreement. Figure 9 shows examples of the comparison between experimental I–V spectra and theoretical calculations. The term dynamical stems from the studies of X-ray diffraction and describes the situation where the response of the crystal to an incident wave is included self-consistently and multiple scattering can occur. The aim of any dynamical LEED theory is to calculate

1911-416: Is controlled by the incident electron energy. From the knowledge of the reciprocal lattice models for the real space lattice can be constructed and the surface can be characterized at least qualitatively in terms of the surface periodicity and the point group. Figure 7 shows a model of an unreconstructed (100) face of a simple cubic crystal and the expected LEED pattern. Since these patterns can be inferred from

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2002-435: Is crucial in various fields, including metallurgy, geology, and materials science. Advancements in crystallographic techniques, such as electron diffraction and X-ray crystallography, continue to expand our understanding of material behavior at the atomic level. In another example, iron transforms from a body-centered cubic (bcc) structure called ferrite to a face-centered cubic (fcc) structure called austenite when it

2093-433: Is flattened by annealing at high temperatures. Once a clean and well-defined surface is prepared, monolayers can be adsorbed on the surface by exposing it to a gas consisting of the desired adsorbate atoms or molecules. Often the annealing process will let bulk impurities diffuse to the surface and therefore give rise to a re-contamination after each cleaning cycle. The problem is that impurities that adsorb without changing

2184-403: Is heated. The fcc structure is a close-packed structure unlike the bcc structure; thus the volume of the iron decreases when this transformation occurs. Crystallography is useful in phase identification. When manufacturing or using a material, it is generally desirable to know what compounds and what phases are present in the material, as their composition, structure and proportions will influence

2275-393: Is the imaginary part of the electron self-energy. In general, R p ≤ 0.2 {\displaystyle R_{p}\leq 0.2} is considered as a good agreement, R p ≃ 0.3 {\displaystyle R_{p}\simeq 0.3} is considered mediocre and R p ≃ 0.5 {\displaystyle R_{p}\simeq 0.5}

2366-403: Is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements . There are 32 possible crystal classes. Each one can be classified into one of

2457-470: Is the primary method for determining the molecular conformations of biological macromolecules , particularly protein and nucleic acids such as DNA and RNA . The double-helical structure of DNA was deduced from crystallographic data. The first crystal structure of a macromolecule was solved in 1958, a three-dimensional model of the myoglobin molecule obtained by X-ray analysis. The Protein Data Bank (PDB)

2548-408: Is then the sphere with radius | k i | {\displaystyle |\mathbf {k} _{i}|} and origin at the center of the incident wave vector. By construction, every wave vector centered at the origin and terminating at an intersection between a rod and the sphere will then satisfy the 2D Laue condition and thus represent an allowed diffracted beam. Figure 4 shows

2639-415: Is tuned to the second harmonic to detect the second derivative. A modern data acquisition system usually contains a CCD/CMOS camera pointed to the screen for diffraction pattern visualization and a computer for data recording and further analysis. More expensive instruments have in-vacuum position sensitive electron detectors that measure the current directly, which helps in the quantitative I–V analysis of

2730-470: Is used by materials scientists to characterize different materials. In single crystals, the effects of the crystalline arrangement of atoms is often easy to see macroscopically because the natural shapes of crystals reflect the atomic structure. In addition, physical properties are often controlled by crystalline defects. The understanding of crystal structures is an important prerequisite for understanding crystallographic defects . Most materials do not occur as

2821-1092: The diffraction patterns of a sample targeted by a beam of some type. X-rays are most commonly used; other beams used include electrons or neutrons . Crystallographers often explicitly state the type of beam used, as in the terms X-ray diffraction , neutron diffraction and electron diffraction . These three types of radiation interact with the specimen in different ways. It is hard to focus x-rays or neutrons, but since electrons are charged they can be focused and are used in electron microscope to produce magnified images. There are many ways that transmission electron microscopy and related techniques such as scanning transmission electron microscopy , high-resolution electron microscopy can be used to obtain images with in many cases atomic resolution from which crystallographic information can be obtained. There are also other methods such as low-energy electron diffraction , low-energy electron microscopy and reflection high-energy electron diffraction which can be used to obtain crystallographic information about surfaces. Crystallography

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2912-591: The trigonal crystal system ), orthorhombic , monoclinic and triclinic . Bravais lattices , also referred to as space lattices , describe the geometric arrangement of the lattice points, and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including quasicrystals , fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above. The crystal structure consists of

3003-400: The 20th century, the study of crystals was based on physical measurements of their geometry using a goniometer . This involved measuring the angles of crystal faces relative to each other and to theoretical reference axes (crystallographic axes), and establishing the symmetry of the crystal in question. The position in 3D space of each crystal face is plotted on a stereographic net such as

3094-537: The 20th century, with the developments of customized instruments and phasing algorithms . Nowadays, crystallography is an interdisciplinary field , supporting theoretical and experimental discoveries in various domains. Modern-day scientific instruments for crystallography vary from laboratory-sized equipment, such as diffractometers and electron microscopes , to dedicated large facilities, such as photoinjectors , synchrotron light sources and free-electron lasers . Crystallographic methods depend mainly on analysis of

3185-517: The 2D form: where a ∗ {\displaystyle \mathbf {a} ^{*}} and b ∗ {\displaystyle \mathbf {b} ^{*}} are the primitive translation vectors of the 2D reciprocal lattice of the surface and k f ∥ {\displaystyle {\textbf {k}}_{f}^{\parallel }} , k i ∥ {\displaystyle {\textbf {k}}_{i}^{\parallel }} denote

3276-460: The Davisson and Germer discovery into the 1970s. In the early 1960s LEED experienced a renaissance, as ultra-high vacuum became widely available, and the post acceleration detection method was introduced by Germer and his coworkers at Bell Labs using a flat phosphor screen. Using this technique, diffracted electrons were accelerated to high energies to produce clear and visible diffraction patterns on

3367-453: The Ewald's sphere for the case of normal incidence of the primary electron beam, as would be the case in an actual LEED setup. It is apparent that the pattern observed on the fluorescent screen is a direct picture of the reciprocal lattice of the surface. The spots are indexed according to the values of h and k . The size of the Ewald's sphere and hence the number of diffraction spots on the screen

3458-418: The I–V curves of a large set of trial structures. A real surface is not perfectly periodic but has many imperfections in the form of dislocations, atomic steps, terraces and the presence of unwanted adsorbed atoms. This departure from a perfect surface leads to a broadening of the diffraction spots and adds to the background intensity in the LEED pattern. SPA-LEED is a technique where the profile and shape of

3549-498: The Miller indices ( ℓmn ) and [ ℓmn ] both simply denote normals/directions in Cartesian coordinates . For cubic crystals with lattice constant a , the spacing d between adjacent (ℓmn) lattice planes is (from above): Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes: For face-centered cubic (fcc) and body-centered cubic (bcc) lattices,

3640-420: The average domain size on the surface and hence the LEED pattern might be a superposition of diffraction beams from domains oriented along different axes of the substrate lattice. However, since the average domain size is generally larger than the coherence length of the probing electrons, interference between electrons scattered from different domains can be neglected. Therefore, the total LEED pattern emerges as

3731-406: The basic symmetry of the surface, cannot easily be identified in the diffraction pattern. Therefore, in many LEED experiments Auger electron spectroscopy is used to accurately determine the purity of the sample. LEED optics is in some instruments also used for Auger electron spectroscopy . To improve the measured signal, the gate voltage is scanned in a linear ramp. An RC circuit serves to derive

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3822-417: The cell edges, measured from a reference point. It is thus only necessary to report the coordinates of a smallest asymmetric subset of particles, called the crystallographic asymmetric unit. The asymmetric unit may be chosen so that it occupies the smallest physical space, which means that not all particles need to be physically located inside the boundaries given by the lattice parameters. All other particles of

3913-436: The component of respectively the reflected and incident wave vector parallel to the sample surface. a ∗ {\displaystyle {\textbf {a}}^{*}} and b ∗ {\displaystyle {\textbf {b}}^{*}} are related to the real space surface lattice, with n ^ {\displaystyle {\hat {\mathbf {n} }}} as

4004-410: The concept of space groups . All possible symmetric arrangements of particles in three-dimensional space may be described by 230 space groups. The crystal structure and symmetry play a critical role in determining many physical properties, such as cleavage , electronic band structure , and optical transparency . Crystal structure is described in terms of the geometry of arrangement of particles in

4095-499: The condition for constructive interference and hence diffraction of scattered electron waves is given by the Laue condition : where ( h , k , l ) is a set of integers, and is a vector of the reciprocal lattice. Note that these vectors specify the Fourier components of charge density in the reciprocal (momentum) space, and that the incoming electrons scatter at these density modulations within

4186-412: The contribution of deeper atoms to the diffraction progressively decreases. Kinematic diffraction is defined as the situation where electrons impinging on a well-ordered crystal surface are elastically scattered only once by that surface. In the theory the electron beam is represented by a plane wave with a wavelength given by the de Broglie hypothesis : The interaction between the scatterers present in

4277-426: The crystal lattice leaves it unchanged. All crystals have translational symmetry in three directions, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration; the crystal has twofold rotational symmetry about this axis. In addition to rotational symmetry, a crystal may have symmetry in

4368-419: The crystal lattice. The magnitudes of the wave vectors are unchanged, i.e. | k f | = | k i | {\displaystyle |\mathbf {k} _{f}|=|\mathbf {k} _{i}|} , because only elastic scattering is considered. Since the mean free path of low-energy electrons in a crystal is only a few angstroms, only the first few atomic layers contribute to

4459-464: The crystal structure of the bulk crystal, known from other more quantitative diffraction techniques, LEED is more interesting in the cases where the surface layers of a material reconstruct, or where surface adsorbates form their own superstructures. Overlaying superstructures on a substrate surface may introduce additional spots in the known (1×1) arrangement. These are known as extra spots or super spots . Figure 6 shows many such spots appearing after

4550-424: The crystal then gives the desired intensities. A common approach in LEED calculations is to describe the scattering potential of the crystal by a "muffin tin" model, where the crystal potential can be imagined being divided up by non-overlapping spheres centered at each atom such that the potential has a spherically symmetric form inside the spheres and is constant everywhere else. The choice of this potential reduces

4641-458: The crystallographic planes are geometric planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows: Some directions and planes are defined by symmetry of the crystal system. In monoclinic, trigonal, tetragonal, and hexagonal systems there is one unique axis (sometimes called the principal axis ) which has higher rotational symmetry than

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4732-494: The desired surface crystallographic orientation is initially cut and prepared outside the vacuum chamber. The correct alignment of the crystal can be achieved with the help of X-ray diffraction methods such as Laue diffraction . After being mounted in the UHV chamber the sample is cleaned and flattened. Unwanted surface contaminants are removed by ion sputtering or by chemical processes such as oxidation and reduction cycles. The surface

4823-419: The diffraction spots. The basic reason for the high surface sensitivity of LEED is that for low-energy electrons the interaction between the solid and electrons is especially strong. Upon penetrating the crystal, primary electrons will lose kinetic energy due to inelastic scattering processes such as plasmon and phonon excitations, as well as electron–electron interactions. In cases where the detailed nature of

4914-592: The diffraction theory for X-rays developed by Bragg and Laue earlier. Before the acceptance of the de Broglie hypothesis, diffraction was believed to be an exclusive property of waves. Davisson and Germer published notes of their electron-diffraction experiment result in Nature and in Physical Review in 1927. One month after Davisson and Germer's work appeared, Thompson and Reid published their electron-diffraction work with higher kinetic energy (thousand times higher than

5005-475: The diffraction. This means that there are no diffraction conditions in the direction perpendicular to the sample surface. As a consequence, the reciprocal lattice of a surface is a 2D lattice with rods extending perpendicular from each lattice point. The rods can be pictured as regions where the reciprocal lattice points are infinitely dense. Therefore, in the case of diffraction from a surface the Laue condition reduces to

5096-498: The energy used by Davisson and Germer) in the same journal. Those experiments revealed the wave property of electrons and opened up an era of electron-diffraction study. Though discovered in 1927, low-energy electron diffraction did not become a popular tool for surface analysis until the early 1960s. The main reasons were that monitoring directions and intensities of diffracted beams was a difficult experimental process due to inadequate vacuum techniques and slow detection methods such as

5187-413: The factor 1/ e . While the inelastic scattering processes and consequently the electronic mean free path depend on the energy, it is relatively independent of the material. The mean free path turns out to be minimal (5–10 Å) in the energy range of low-energy electrons (20–200 eV). This effective attenuation means that only a few atomic layers are sampled by the electron beam, and, as a consequence,

5278-424: The following series: This arrangement of atoms in a crystal structure is known as hexagonal close packing (hcp) . If, however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises: This type of structural arrangement is known as cubic close packing (ccp) . The unit cell of

5369-437: The form of mirror planes, and also the so-called compound symmetries, which are a combination of translation and rotation or mirror symmetries. A full classification of a crystal is achieved when all inherent symmetries of the crystal are identified. Lattice systems are a grouping of crystal structures according to the point groups of their lattice. All crystals fall into one of seven lattice systems. They are related to, but not

5460-400: The incoherent sum of the diffraction patterns associated with the individual domains. Figure 8 shows the superposition of the diffraction patterns for the two orthogonal domains (2×1) and (1×2) on a square lattice, i.e. for the case where one structure is just rotated by 90° with respect to the other. The (1×2) structure and the respective LEED pattern are shown in Figure 7. It is apparent that

5551-429: The individual scatters present in the surface, where the effective field is the sum of the primary field and the field emitted from all the other atoms. This must be done in a self-consistent way, since the emitted field of an atom depends on the incident effective field upon it. Once the effective field incident on each atom is determined, the total field emitted from all atoms can be found and its asymptotic value far from

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5642-424: The inelastic processes is unimportant, they are commonly treated by assuming an exponential decay of the primary electron-beam intensity I 0 in the direction of propagation: Here d is the penetration depth, and Λ ( E ) {\displaystyle \Lambda (E)} denotes the inelastic mean free path , defined as the distance an electron can travel before its intensity has decreased by

5733-429: The intensities of diffraction of an electron beam impinging on a surface as accurately as possible. A common method to achieve this is the self-consistent multiple scattering approach. One essential point in this approach is the assumption that the scattering properties of the surface, i.e. of the individual atoms, are known in detail. The main task then reduces to the determination of the effective wave field incident on

5824-529: The intensity of diffraction beam spots is measured. The spots are sensitive to the irregularities in the surface structure and their examination therefore permits more-detailed conclusions about some surface characteristics. Using SPA-LEED may for instance permit a quantitative determination of the surface roughness, terrace sizes, dislocation arrays, surface steps and adsorbates. Although some degree of spot profile analysis can be performed in regular LEED and even LEEM setups, dedicated SPA-LEED setups, which scan

5915-436: The intensity versus incident electron energy. The I–V curves can be recorded by using a camera connected to computer controlled data handling or by direct measurement with a movable Faraday cup. The experimental curves are then compared to computer calculations based on the assumption of a particular model system. The model is changed in an iterative process until a satisfactory agreement between experimental and theoretical curves

6006-423: The local symmetry of the surface structure is twofold while the LEED pattern exhibits a fourfold symmetry. Figure 1 shows a real diffraction pattern of the same situation for the case of a Si(100) surface. However, here the (2×1) structure is formed due to surface reconstruction . The inspection of the LEED pattern gives a qualitative picture of the surface periodicity i.e. the size of the surface unit cell and to

6097-398: The material's properties. Each phase has a characteristic arrangement of atoms. X-ray or neutron diffraction can be used to identify which structures are present in the material, and thus which compounds are present. Crystallography covers the enumeration of the symmetry patterns which can be formed by atoms in a crystal and for this reason is related to group theory . X-ray crystallography

6188-423: The other two axes. The basal plane is the plane perpendicular to the principal axis in these crystal systems. For triclinic, orthorhombic, and cubic crystal systems the axis designation is arbitrary and there is no principal axis. For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a ); similarly for the reciprocal lattice. So, in this common case,

6279-420: The physical properties of individual crystals themselves. Each book is about 1000 pages and the titles of the books are: Crystal structure In crystallography , crystal structure is a description of ordered arrangement of atoms , ions , or molecules in a crystalline material . Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along

6370-421: The platelike particles can slip along each other in the plane of the plates, yet remain strongly connected in the direction perpendicular to the plates. Such mechanisms can be studied by crystallographic texture measurements. Crystallographic studies help elucidate the relationship between a material's structure and its properties, aiding in developing new materials with tailored characteristics. This understanding

6461-423: The possibility of multiple scattering, was established in the late 1960s. With this theory, it later became possible to reproduce experimental data with high precision. In order to keep the studied sample clean and free from unwanted adsorbates, LEED experiments are performed in an ultra-high vacuum environment (residual gas pressure <10  Pa). The main components of a LEED instrument are: The sample of

6552-411: The primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions . The spacing d between adjacent ( hkℓ ) lattice planes is given by: The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on

6643-416: The primitive translation vectors of the superlattice { a s ,  b s } are linked to the primitive translation vectors of the underlying (1×1) lattice { a ,  b } in the following way The matrix for the superstructure then is Similarly, the primitive translation vectors of the lattice describing the extra spots { a s ,  b s } are linked to the primitive translation vectors of

6734-621: The principal directions of three-dimensional space in matter. The smallest group of particles in material that constitutes this repeating pattern is unit cell of the structure. The unit cell completely reflects symmetry and structure of the entire crystal, which is built up by repetitive translation of unit cell along its principal axes. The translation vectors define the nodes of Bravais lattice . The lengths of principal axes/edges, of unit cell and angles between them are lattice constants , also called lattice parameters or cell parameters . The symmetry properties of crystal are described by

6825-503: The problem to scattering from spherical potentials, which can be dealt with effectively. The task is then to solve the Schrödinger equation for an incident electron wave in that "muffin tin" potential. In LEED the exact atomic configuration of a surface is determined by a trial and error process where measured I–V curves are compared to computer-calculated spectra under the assumption of a model structure. From an initial reference structure

6916-410: The reciprocal lattice { a ,  b } G is related to G in the following way An essential problem when considering LEED patterns is the existence of symmetrically equivalent domains. Domains may lead to diffraction patterns that have higher symmetry than the actual surface at hand. The reason is that usually the cross sectional area of the primary electron beam (~1 mm ) is large compared to

7007-482: The same as the seven crystal systems . aP mP mS oP oS oI oF tP tI hR hP cP cI cF The most symmetric, the cubic or isometric system, has the symmetry of a cube , that is, it exhibits four threefold rotational axes oriented at 109.5° (the tetrahedral angle ) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal , tetragonal , rhombohedral (often confused with

7098-520: The same group of atoms, the basis , positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its crystallographic point group . A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to

7189-536: The screen. Ironically the post-acceleration method had already been proposed by Ehrenberg in 1934. In 1962 Lander and colleagues introduced the modern hemispherical screen with associated hemispherical grids. In the mid-1960s, modern LEED systems became commercially available as part of the ultra-high-vacuum instrumentation suite by Varian Associates and triggered an enormous boost of activities in surface science. Notably, future Nobel prize winner Gerhard Ertl started his studies of surface chemistry and catalysis on such

7280-407: The second derivative , which is then amplified and digitized. To reduce the noise, multiple passes are summed up. The first derivative is very large due to the residual capacitive coupling between gate and the anode and may degrade the performance of the circuit. By applying a negative ramp to the screen this can be compensated. It is also possible to add a small sine to the gate. A high-Q RLC circuit

7371-401: The seven crystal systems. In addition to the operations of the point group, the space group of the crystal structure contains translational symmetry operations. These include: There are 230 distinct space groups. By considering the arrangement of atoms relative to each other, their coordination numbers, interatomic distances, types of bonding, etc., it is possible to form a general view of

7462-631: The structure. The APFs and CNs of the most common crystal structures are shown below: The 74% packing efficiency of the FCC and HCP is the maximum density possible in unit cells constructed of spheres of only one size. Interstitial sites refer to the empty spaces in between the atoms in the crystal lattice. These spaces can be filled by oppositely charged ions to form multi-element structures. They can also be filled by impurity atoms or self-interstitials to form interstitial defects . Low-energy electron diffraction Low-energy electron diffraction ( LEED )

7553-447: The structures and alternative ways of visualizing them. The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer were placed directly over plane A, this would give rise to

7644-611: The surface and the incident electrons is most conveniently described in reciprocal space. In three dimensions the primitive reciprocal lattice vectors are related to the real space lattice { a , b , c } in the following way: For an incident electron with wave vector k i = 2 π / λ i {\displaystyle \mathbf {k} _{i}=2\pi /\lambda _{i}} and scattered wave vector k f = 2 π / λ f {\displaystyle \mathbf {k} _{f}=2\pi /\lambda _{f}} ,

7735-405: The surface normal, in the following way: The Laue-condition equation can readily be visualized using the Ewald's sphere construction. Figures 3 and 4 show a simple illustration of this principle: The wave vector k i {\displaystyle \mathbf {k} _{i}} of the incident electron beam is drawn such that it terminates at a reciprocal lattice point. The Ewald's sphere

7826-401: The syntax ( hkℓ ) denotes a plane that intercepts the three points a 1 / h , a 2 / k , and a 3 / ℓ , or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it means that the planes do not intersect that axis (i.e., the intercept

7917-404: The tools of X-ray crystallography can convert into detailed positions of atoms, and sometimes electron density. At larger scales it includes experimental tools such as orientational imaging to examine the relative orientations at the grain boundary in materials. Crystallography plays a key role in many areas of biology, chemistry, and physics, as well new developments in these fields. Before

8008-420: The unit cell are generated by the symmetry operations that characterize the symmetry of the unit cell. The collection of symmetry operations of the unit cell is expressed formally as the space group of the crystal structure. Vectors and planes in a crystal lattice are described by the three-value Miller index notation. This syntax uses the indices h , k , and ℓ as directional parameters. By definition,

8099-445: The unit cells. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure. The geometry of the unit cell is defined as a parallelepiped , providing six lattice parameters taken as the lengths of the cell edges ( a , b , c ) and the angles between them (α, β, γ). The positions of particles inside the unit cell are described by the fractional coordinates ( x i , y i , z i ) along

8190-462: Was conducted in 1912 by Max von Laue , while electron diffraction was first realized in 1927 in the Davisson–Germer experiment and parallel work by George Paget Thomson and Alexander Reid. These developed into the two main branches of crystallography, X-ray crystallography and electron diffraction. The quality and throughput of solving crystal structures greatly improved in the second half of

8281-406: Was established as an ubiquitous tool in surface science only with the advances in vacuum generation and electron detection techniques. The theoretical possibility of the occurrence of electron diffraction first emerged in 1924, when Louis de Broglie introduced wave mechanics and proposed the wavelike nature of all particles. In his Nobel-laureated work de Broglie postulated that the wavelength of

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