Unit Cells

6 Key excerpts on "Unit Cells"

• 

  eBook - ePub

  An Introduction to Nuclear Materials

  Fundamentals and Applications

  • K. Linga Murty, Indrajit Charit(Authors)
  • 2013(Publication Date)
  • Wiley-VCH

    (Publisher)

  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. They are summarized in Table 2.1

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introduction to Nanoscience

  • Gabor L. Hornyak, Joydeep Dutta, H.F. Tibbals, Anil Rao(Authors)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  262 Introduction to Nanoscience structural planes of most crystalline materials ( Fig. 5.9 ). Crystal structure is based on regularly repeating elements that form a pattern in three dimensions (e.g., the unit cell). The unit cell is described by a set of lattice parameters and is the irreduc-ible representation of the crystal structure. For example, knowledge of lattice con-stants such as unit cell edge length ( a,b,c ) and crystallographic axis angle ( a , b , g ) allows for facilitative quantification of material structure. In other words, translation of structurally equivalent positions of the unit cell over and over again results in a material with long-range order—ultimately, a crystal. In contrast, such long-range order is absent in amorphous materials such as glass, liquids, and gases. Calculate the number of volume atoms, the number of surface atoms, and the percentage of surface atoms N s to volume atoms N v for a spherical cluster of N a atoms. Relate the cluster radius ( R c ), surface area ( S c ), and volume ( V c ) to the radius ( R a ), surface area ( S a ), and volume ( V a ) of an individual atom in the cluster [15].

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Selection of Engineering Materials and Adhesives

  • P.E. Fisher, P.E., Lawrence W. Fisher(Authors)
  • 2005(Publication Date)
  • CRC Press

    (Publisher)

  Crystal Systems The following four are considered basic Unit Cells: ● Simple ● Body-Centered Selection of Engineering Materials and Adhesives 36 Figure 2.9 Unit cell. These repeating units of atoms or molecules are represented by a specific set of axis, edge lengths, and faces to define a crystallographic structure. ● Face-Centered ● Base-Centered There are three variations possible in the cubic lattice system: simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC). The tetragonal system occurs in simple and body-centered variations. The monoclinic system occurs in simple and base-centered types. The orthorhombic system occurs in all four types. Rhombohedral, hexagonal, and triclinic systems occur only in one variation each. Crystallographers have shown, by varying the unit cell’s axial lengths and angles, that only seven crystal systems are necessary to define all existing crystal structures ( Table 2.4 ). Having defined the seven basic crystal systems the question remains as to how many unit cell arrangements can be formed. A.J. Bravais determined that 14 standard Unit Cells could describe all possible lattice networks ( Figure 2.11 ). Miller Indices and Crystallographic Planes Even in the simplest of structures there is a need for a common language to describe specific points, directions, and planes in crystals. This need is filled by defining any desired point, direc-tion, or plane by means of atomic positions, direction indices, and Miller indices for planes. Miller indices are defined as the reciprocals of the intercepts made by the planes on the crystal axes, the x -, y - and z -axis. 37 Structure of Materials b Y a c Figure 2.10 Size and shape of the unit cell. The cell is also described by interaxial angles defining surface orientation. Lattice Positions The position of an atom in a unit cell is described by using a rectangular x , y , z coordinate system which follows the right hand rule ( Figure 2.12 ).

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Materials Science and Engineering

  An Introduction

  • William D. Callister, Jr., David G. Rethwisch(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Sometimes the term lattice is used in the context of crystal structures; in this sense lattice means a three-dimensional array of points coinciding with atom posi- tions (or sphere centers). crystal structure lattice (a) (b) (c) Figure 3.1 For the face-centered cubic crystal structure, (a) a hard-sphere unit cell representation, (b) a reduced- sphere unit cell, and (c) an aggregate of many atoms. [Figure (c) adapted from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. I, Structure, John Wiley & Sons, 1964. Reproduced with permission of Janet M. Moffatt.] The atomic order in crystalline solids indicates that small groups of atoms form a repeti- tive pattern. Thus, in describing crystal structures, it is often convenient to subdivide the structure into small repeat entities called Unit Cells. Unit Cells for most crystal structures are parallelepipeds or prisms having three sets of parallel faces; one is drawn within the aggregate of spheres (Figure 3.1c), which in this case happens to be a cube. A unit cell is unit cell 3.3 Unit Cells 3.4 Metallic Crystal Structures • 51 chosen to represent the symmetry of the crystal structure, wherein all the atom positions in the crystal may be generated by translations of the unit cell integral distances along each of its edges. Thus, the unit cell is the basic structural unit or building block of the crystal structure and defines the crystal structure by virtue of its geometry and the atom positions within. Convenience usually dictates that parallelepiped corners coincide with centers of the hard-sphere atoms. Furthermore, more than a single unit cell may be chosen for a particular crystal structure; however, we generally use the unit cell having the high- est level of geometrical symmetry. The atomic bonding in this group of materials is metallic and thus nondirectional in nature.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Callister's Materials Science and Engineering

  • William D. Callister, Jr., David G. Rethwisch(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Sometimes the term lattice is used in the context of crystal structures; in this sense lattice means a three-dimensional array of points coinciding with atom posi- tions (or sphere centers). crystal structure lattice (a) (b) (c) Figure 3.1 For the face-centered cubic crystal structure, (a) a hard-sphere unit cell representation, (b) a reduced- sphere unit cell, and (c) an aggregate of many atoms. [Figure (c) adapted from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. I, Structure, John Wiley & Sons, 1964. Reproduced with permission of Janet M. Moffatt.] The atomic order in crystalline solids indicates that small groups of atoms form a repeti- tive pattern. Thus, in describing crystal structures, it is often convenient to subdivide the structure into small repeat entities called Unit Cells. Unit Cells for most crystal structures are parallelepipeds or prisms having three sets of parallel faces; one is drawn within the aggregate of spheres (Figure 3.1c), which in this case happens to be a cube. A unit cell is unit cell 3.3 Unit Cells 52 • Chapter 3 / The Structure of Crystalline Solids chosen to represent the symmetry of the crystal structure, wherein all the atom positions in the crystal may be generated by translations of the unit cell integral distances along each of its edges. Thus, the unit cell is the basic structural unit or building block of the crystal structure and defines the crystal structure by virtue of its geometry and the atom positions within. Convenience usually dictates that parallelepiped corners coincide with centers of the hard-sphere atoms. Furthermore, more than a single unit cell may be chosen for a particular crystal structure; however, we generally use the unit cell having the high- est level of geometrical symmetry. The atomic bonding in this group of materials is metallic and thus nondirectional in nature.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Characterization of Condensed Matter

  An Introduction to Composition, Microstructure, and Surface Methods

  • Yujun Song, Qingwei Liao(Authors)
  • 2023(Publication Date)
  • Wiley-VCH

    (Publisher)

  In mineralogy and crystallography, crystal structure describes a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long‐range order and a characteristic symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called Unit Cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. A crystalline structure is similar to a kind of three‐dimensional wallpaper, an infinite repetition of some entity (such as a series of atoms or molecules). The procedure for creating a pattern consists of performing the point group operations that define it, such as rotation, reflection, and inversion. Making the wallpaper requires a movement (with or without rotation and reflection) to create a lattice that is a complete structure. In practice, crystalline structures can consist of single lattices or multiple grids to synthesize complex crystalline molecules. As long as it is repeatable, X‐ray diffraction can be used to determine its structure.

  3.2.3 Unit Cell and Unit Vectors

  The unit cell (the smallest repeating entity) is the basic structural unit of a crystal structure. Its geometry and atomic positions define the crystal structure, representing the symmetry of the crystal structure (Figure 3.6 ). More than one unit cell can usually be chosen for a given crystal, but by convention/convenience, the one with the highest symmetry is selected. Some metals, as well as nonmetals, may assume more than one crystal structure, a phenomenon known as polymorphism. In elemental solids, the condition is often termed allotropy. The prevailing crystal structure depends on conditions – mainly temperature and pressure. Since the unit cell of any crystal can be regarded as a parallelepiped, what is the difference between different crystals? There are mainly two differences: (i) different crystal Unit Cells may have different sizes and shapes; (ii) the type, number, and distribution of atoms surrounding around each lattice point may be different.

  Figure 3.6

  Lattice and crystal structures of γ‐Fe and Cu3 Au.

  The size of the unit cell obviously depends on the length of the three crystal axes a, b, and c, while the shape of a unit cell depends on the angles α, β, γ between them (Figure 3.7 ). The parameters a, b, c, α, β and γ are called the lattice parameters of a crystal structure. In general, a ≠ b ≠ c and α ≠ β ≠ γ. The six lattice parameters (or three lattice vectors a, b, and c) describe the shape and size of the unit cell and determine the entire lattice that are formed by the translation of these vectors. That is to say, any lattice point in the spatial lattice can be generated by repeated translation along the vectors a, b, and c

  Sign up to read

  Learn more about book