3D Lattice
A 3D lattice refers to a regular, repeating arrangement of points or nodes in three-dimensional space. In physics, this concept is often used to model the structure of crystalline solids, where atoms or molecules are arranged in a repeating pattern. The lattice structure is important for understanding the properties and behavior of materials at the atomic level.
7 Key excerpts on "3D Lattice"
eBook - ePubAn Introduction to Nuclear Materials
Fundamentals and Applications
- K. Linga Murty, Indrajit Charit(Authors)
- 2013(Publication Date)
- Wiley-VCH(Publisher)
...This is exemplified by the materials science tetrahedron, as depicted in Figure 2.2. Figure 2.2 Materials science and engineering tetrahedron. This theme is equally applicable to the nuclear materials. For example, materials scientists and engineers study microstructural features (grain size, type of second phases and their relative proportions, grain boundary character distribution to name a few) to elucidate the behavior of a material. These are structural features that are influenced by the nature of the processing techniques (casting, rolling, forging, powder metallurgy, and so forth) employed, leading to changes in properties. This understanding will be very helpful as we wade through the subsequent chapters. A lattice is an array of points in three dimensions such that each point has identical surroundings. When such lattice point is assigned one or more atoms/ions (i.e., basis), a crystal is formed. In this chapter, we present a simple treatment of the crystal structure and relate it to its importance with respect to nuclear materials. There are 7 basic crystal systems and a total of 14 unique crystal structures (Bravais lattices) that can be found in most elemental solids. These are based on the crystal symmetry and the arrangement of atoms as described in the following sections. 2.1.1 Unit Cell A unit cell is the smallest building block of a crystal, which when repeated in translation (i.e., with no rotation) in three-dimension can create a single crystal. Therefore, a single crystal or a “grain” in a polycrystalline material would contain many of these unit cells. A general unit cell can be created based on three lattice translation vectors (a, b, and c) on three orthogonal axes and interaxial angles (α, β, and γ), which are also known as lattice parameters or lattice constants. Figure 2.3 illustrates the definitions of the lattice parameters and the angles. There are seven basic crystal systems...
eBook - ePub- Alexander McPherson(Author)
- 2011(Publication Date)
- Wiley-Blackwell(Publisher)
...A lattice, mathematically, is a discrete, discontinuous function. A lattice is absolutely zero everywhere except at very specific, predictable, periodically distributed points where it takes on a value of one. We can begin to see, from the discussion above that a crystal may also be thought of as the combination of two things, a cluster of real molecules whose atoms are related by space group symmetry and an imaginary point lattice that describes their periodic placement in three-dimensional space. The lattice, notice, defines the unit cell, just as the unit cell defines the lattice. FIGURE 3.8 Because of translational periodicity of unit cells in a crystal, the unit cell edges may be defined as having lengths of 1 (period along, 1 period along, and 1 period along). That is, the unit cell edges define the unitary vectors of the crystal lattice. This is consistent with the idea of the unit cell contents constituting a periodic, three-dimensional electron density wave traveling along the directions of the unit cell vectors,, and. Points in the unit cell, such as the two indicated here, can then be defined in terms of three fractions (of the three periods). Addition of 1 to a fractional coordinate simply yields the identical point in an adjacent unit cell. Addition of integers n, m, p to any fractional coordinate generates the identical point n, m and p unit cells away along,, and, respectively. The mathematical equations and expressions dealing with crystals will always be formulated in terms of this fractional coordinate system. Indeed any atom in any unit cell, anywhere in the crystal, can always be defined in terms of this fractional coordinate system. FIGURE 3.9 If any arbitrary point is sequentially translated along direction by m | |, where m is integral, then a periodic, linear array of points is generated. If that line of points is then sequentially translated along by n | |, a periodic two-dimensional plane of points results...
eBook - ePubNanotechnology
Synthesis to Applications
- Sunipa Roy, Chandan Kumar Ghosh, Chandan Kumar Sarkar, Sunipa Roy, Chandan Kumar Ghosh, Chandan Kumar Sarkar(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...The lattice point is an imaginary, mathematical concept. A group of atoms or molecules known as basis arrange themselves in a unique manner around the lattice points. A unit cell is the smallest volume of a crystal structure having the same symmetry as the whole crystal. The unit cell when repeated along all directions forms the crystal. A unit cell is known as primitive when it has only one lattice point and is known as nonprimitive when it has more than one lattice point (Figure 3.1). FIGURE 3.1 Crystal. The widely used extrinsic semiconductors in the electronic industry are formed by doping of crystal structures. We create free holes (in case of p type) and free electrons (in case of n type) by means of doping such that they can help in conduction. Silicon, germanium, and gallium are widely used in semiconductors. Thus, the crystal structure plays an important role in the creation of semiconductors, which are the basic building blocks of diodes and transistors. 3.3 Energy Bands We know that the energy of the bound electron in an atom is quantized. It occupies atomic orbitals of discrete energy levels. As the atoms combine to form molecules, the atomic orbitals overlap. According to Pauli’s exclusion principle, no two electrons can have the same quantum number in a molecule. In the case of crystal lattices, a large number of identical atoms combine to form a molecule. Here the atomic orbitals split into different energy levels. A large number of closely spaced energy levels are formed. These closely spaced energy levels are known as energy bands. The finite energy gap between two energy bands is known as band gap (Figure 3.2). FIGURE 3.2 Splitting of energy bands. Band structure can be found by solving Schrödinger’s equation, which gives Bloch waves as the solution, of the form (3.1) (3.1) ψ x = u x e i k x where k is the wave vector or constant of motion. The energy E has discontinuities with forbidden gaps for the particles...
eBook - ePubGateway to Condensed Matter Physics and Molecular Biophysics
Concepts and Theoretical Perspectives
- Ranjan Chaudhury(Author)
- 2021(Publication Date)
- Apple Academic Press(Publisher)
...CHAPTER 3 Lattice Dynamics, Phonons, and Some Applications for Optical Properties of Solids 3.1 INTRODUCTION A large part of condensed matter physics, both theoretical and experimental, deals with studies of crystal lattices, directly or indirectly. In particular, both crystal structure (i.e., lattice geometry) and the lattice dynamics can play sometimes very cmcial roles in deciding the properties of various condensed matter systems and also the occurrences of different phenomena like ferroelectricity, superconductivity, and magnetism. There are also extremely interesting and exciting phenomena in solids involving various ‘optoelectronic,’ i.e., occurring with the combination of optical and electronic components processes. The occurrences of these processes in low-dimensional systems have found tremendous technological applications. As explained detail in Chapter 1, the lattice of atoms or more precisely the ion cores provide the Bloch potential essential for the electronic energy bands to form in a regular solid. The ion cores themselves interact with each other to form a stable and well-defined lattice. This inter-atomic or inter-ionic potential is attractive at large spatial separation and repulsive at shorter distances. This is often modeled by Van-der-Waals or Lennard-Jones type of potential functions (Huang, 1975). Both of these functions exhibit the required spatial dependence of the inter-atomic potential. They arise from the interactions involving the nuclei and the electron clouds of the neighboring atoms. They have contributions from dipole-dipole and also higher-order multipole-multipole interactions in general. The attractive-repulsive nature of this inter-atomic interaction leads to the existence of an equilibrium distance between the near neighbor atoms. This is identified with the lattice parameter of the corresponding crystal lattice...
eBook - ePubApplied Welding Engineering
Processes, Codes, and Standards
- Ramesh Singh(Author)
- 2011(Publication Date)
- Butterworth-Heinemann(Publisher)
...Chapter 3. Physical Metallurgy Chapter Outline Crystal Lattices 13 Crystal Structure Nomenclature 14 Solidification 14 Lever Rule of Solidification 14 Constitutional Supercooling 16 Elementary Theory of Nucleation 17 Allotropy 18 Crystal Imperfections 21 Grain Size 21 In the solid state materials have a crystal structure, broadly defined as the arrangement of atoms or molecules. The arrangement at the atomic and molecular level is called the microstructure of the material, and includes abnormalities in the crystalline structure, if any are present. Seven different systems of crystal are known, and are named by the Pearson symbols. Phase equilibria are important for crystalline systems, particularly those governing their solidification properties. In alloys there is unlimited solid solubility in both the liquid and solid phase, and they freeze or melt over a range of temperature, rather than having a fixed melting point. This behaviour is shown by a phase diagram. Keywords crystal structure, Pearson symbols, phase equilibrium, phase diagram, solidification. In the previous chapters we briefly introduced the alloying process and various alloys of steel, and so in this chapter we turn to the fundamental physics of metallurgy. In the solid state materials have a crystal structure, broadly defined as the arrangement of atoms or molecules. The arrangement at the atomic and molecular level is collectively called the microstructure of material. The arrangement may include the abnormalities in the crystalline structure. Crystal Lattices The three-dimensional network of imaginary lines connecting the atoms in a regular solid structure is called the space lattice. A crystal is an arrangement in three dimensions of atoms or molecules in a repetitive pattern. The smallest unit that possesses the full symmetry of the crystal is called the unit cell, the edges of which form three axes, a, b and c...
eBook - ePub- Jeffrey Gaffney, Nancy Marley(Authors)
- 2017(Publication Date)
- Elsevier(Publisher)
...used to relieve internal stresses in a glass by slowly heating the glass to a temperature just below the softening point, maintaining the temperature for a period of time, and allowing it to cool slowly. Atomic solids solids composed of atoms connected by covalent bonds. Band gap (E g) the energy region between the valence band and the conductance band in a semiconductor. Born-Haber cycle an application of Hess’s law used to calculate lattice energies. Cleaving causing an ionic solid to split along the axis parallel to the planes of the ions by an application of an outside force. Close packed structure the most tightly packed and space-efficient arrangement of ions in a crystal lattice. Conduction band a group of molecular orbitals of the same energy in a crystal lattice through which electrons are free to move easily. Coordination number the number of nearest neighbor ions of opposite charge surrounding any ion in a crystal structure. Crystal lattice the simplest repeating unit in a crystalline solid. Crystalline solid a solid material with the constituent species arranged in a highly ordered microscopic structure that extends in all directions. Crystallization the process by which a crystalline solid is formed. Delocalized electrons electrons in a molecule, ion, or solid metal that are not associated with a single atom or a covalent...
eBook - ePubRadiation Detection
Concepts, Methods, and Devices
- Douglas McGregor, J. Kenneth Shultis(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
...The lattice is the structure onto which the basis is projected and repeated. Together they form a crystal structure. Figure 12.1 depicts the basis and lattice of a crystal. When the basis is placed upon each of the lattice points, a crystal is formed. The lattice itself is a periodic structure that can be reduced to a single unit cell which, when repeated, reproduces the entire crystal. In other words, by simply repeating the unit cell over and over with each unit cell placed adjacent to the last, a crystal is formed. Figure 12.2 shows two possible unit cells for the lattice, both of which are valid. Unit cells do not have to be primitive cells, which are the smallest possible unit cell that, when repeated, forms the crystal. In fact, unit cells are not unique and can be formed in many geometric shapes, provided that the shapes, when stacked, again form the crystal. Figure 12.1. A crystal is formed by repeating a basis over the points of a lattice. Figure 12.2. Two possible unit cells for the lattice shown. The unit cell is related to the lattice in terms of basis vectors. Consider the two lattices shown in Figure 12.3. Then a is a basis unit vector of length a, and b is a basis unit vector of length b. An equivalent point anywhere on the lattice can be found by translating the basis vectors by integers. In other words, equivalent points in a two-dimensional lattice can be represented by r = h a + k b, (12.1) where h and k are positive or negative integers. As a result, the entire crystal can be reproduced by translating the unit cell by allowing the integers h and k to assume all possible values. Figure 12.3. The unit cell can be defined by the unit basis vectors, a and b with magnitudes a and b, respectively. Shown are (a) rectangular and (b) oblique 2-dimensional lattices...
