Simple Cubic Unit Cell

12 Key excerpts on "Simple Cubic Unit Cell"

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  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.

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  Materials Science and Engineering, P-eBK

  An Introduction

  • William D. Callister, Jr., David G. Rethwisch, Aaron Blicblau, Kiara Bruggeman, Michael Cortie, John Long, Judy Hart, Ross Marceau, Ryan Mitchell, Reza Parvizi, David Rubin De Celis Leal, Steven Babaniaris, Subrat Das, Thomas Dorin, Ajay Mahato, Julius Orwa(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  FIGURE 3.1 For the face‐centred cubic crystal structure, (a) a hard‐sphere unit cell representation, (b) a reduced‐sphere unit cell, and (c) an aggregate of many atoms. (a) (b) (c) Source: 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. 3.3 Unit cells The atomic order in crystalline solids indicates that small groups of atoms form a repetitive 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 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 centres 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 highest level of geometrical symmetry. 3.4 Metallic crystal structures The atomic bonding in this group of materials is metallic and thus nondirectional in nature. Consequently, there are minimal restrictions as to the number and position of nearest‐neighbour atoms; this leads to relatively large numbers of nearest neighbours and dense atomic packings for most metallic crystal structures.

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  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.

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  Engineering Chemistry

  Fundamentals and Applications

  • Shikha Agarwal(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  604 Engineering Chemistry: Fundamentals and Applications Triclinic a ≠ b ≠ c a ≠ b ≠ g ≠ 90° Hexagonal a = b ≠ c a = b = 90° g = 120° Rhombohedral (or trigonal) a = b = c a = b = g ≠ 90° Figure 11.4 Fourteen Bravais lattices 11.5 Types of Unit Cells and Number of Atoms per Unit Cell There are three types of unit cells for cubic crystals – simple cubic, body centred cubic and face centred cubic. (i) Simple cubic In a simple cubic arrangement, the atoms, ions or molecules are present only at the corners of the cube (Fig. 11.5 a). The atom present at each corner is shared by 8 cubes and hence it contributes only 1/8 to each unit cell (Fig. 11.5 b). The total number of atoms per unit cell in a simple cubic arrangement is 8 corner atoms × 1/8 atom per unit cell = 1 atom Thus, a Simple Cubic Unit Cell has one atom per unit cell. Only one metal, polonium has a Simple Cubic Unit Cell. Figure 11.5 (a) Simple cubic arrangement. (b) Number of spheres per unit cell Solid State 605 (ii) Body centred cubic A body centred cubic (BCC) has atoms, ions or molecules at all the corners as well as at the centre of the cube (Fig. 11.6 a). Each corner atom contributes 1/8 to the unit cell; the atom at the centre belongs exclusively to the unit cell and is not shared by any other cube (Fig. 11.6 b). The total number of atoms per unit cell in a BCC arrangement can be determined as follows. Figure 11.6 (a) Body centred cubic arrangement. (b) Number of spheres per unit cell 8 corner atoms × 1/8 atom per unit cell = 1 atom Number of atoms at the centre of the cube = 1 \ Total number of atoms per unit cell in a BCC arrangement = 1 + 1 = 2 Thus, a body centred cube has two atoms per unit cell. (iii) Face centred cubic A face centred cubic arrangement (FCC) is also termed as a cubic closed packed (CCP) arrangement. Apart from having atoms, ions or molecules at the corners, a face centred cube has atoms at the centre of each of the six faces (Fig. 11.7 a).

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  Introductory Solid State Physics

  • H.P. Myers(Author)
  • 1997(Publication Date)
  • CRC Press

    (Publisher)

  Simple crystal structures have bases containing only a few atoms, but it is possible, particularly in biological materials, to find many hundreds of atoms in the basis. By way of illustration we consider the body-centred cubic array (Fig. 2.6). This is a true Bravais lattice, but the accepted unit cell may be considered as built up from a primitive simple cubic cell to which has been added a basis, namely one atom at a corner and one at the centre of the cube. Thus, although the arrangement of points is part of a true lattice, we treat it as a structure derived from the simple cubic lattice. An example of a more complicated structure built around the body-centred cubic lattice is that of α-Mn. Although the lattice is body-centred cubic, the unit cell contains 58 atoms, the basis being a group of 29 Mn atoms. Geometrically the introduction of the basis, being an assembly of points associated with each and every lattice point, introduces the possibility for new symmetry elements such as rotations and reflections of the basis about axes and planes through the associated lattice point, each operation or combination of operations turning the arrangement into itself again. Crystallography 31 Figure 2.3 The Wigner-Seitz cell that surrounds each lattice point. Figure 2.4 The 14 Bravais space lattices; they may be described in terms of suitable primitive cells. However, it is more convenient and conventional to use a larger unit cell which often involves atoms in end, body-centre or face-centre positions. This procedure also has the advantage of allowing orthogonal axes, (a), (b), (c) cubic systems; (d), (e) tetragonal systems; (f), (g), (h), (i) orthorhombic systems; (j), (k) rhombohedral systems; (l), (m), (n) monoclinic and triclinic systems. Inspection shows that translational symmetry is incompatible with 5-fold rotational symmetry.

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  Engineering Chemistry

  Fundamentals and Applications

  • Shikha Agarwal(Author)
  • 2016(Publication Date)
  • Cambridge University Press

    (Publisher)

  (i) Simple cubic In a simple cubic arrangement, the atoms, ions or molecules are present only at the corners of the cube (Fig. 9.5(a)). The atom present at each corner is shared by 8 cubes and hence it contributes only 1/8 to each unit cell (Fig. 9.5(b)). The total number of atoms per unit cell in a simple cubic arrangement is 8 corner atoms × 1/8 atom per unit cell = 1 atom Thus, a Simple Cubic Unit Cell has one atom per unit cell. Only one metal, polonium has a Simple Cubic Unit Cell. Figure 9.5 (a) Simple cubic arrangement (b) number of spheres per unit cell Solid State 485 (ii) Body centred cubic A body centred cubic (BCC) has atoms, ions or molecules at all the corners as well as at the centre of the cube (Fig. 9.6(a)). Each corner atom contributes 1/8 to the unit cell; the atom at the centre belongs exclusively to the unit cell and is not shared by any other cube (Fig. 9.6(b)). The total number of atoms per unit cell in a BCC arrangement can be determined as follows. Figure 9.6 (a) Body centred cubic arrangement (b) number of spheres per unit cell 8 corner atoms × 1/8 atom per unit cell = 1 atom Number of atoms at the centre of the cube = 1 ∴ Total number of atoms per unit cell in a BCC arrangement = 1 + 1 = 2 Thus, a body centred cube has two atoms per unit cell. (iii) Face centred cubic A face centred cubic arrangement (FCC) is also termed as a cubic closed packed (CCP) arrangement. Apart from having atoms, ions or molecules at the corners, a face centred cube has atoms at the centre of each of the six faces (Fig. 9.7(a)). The corner atom contributes 1/8 to each unit cell and the atoms at the faces contribute ½ to each unit cell Fig.

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  Principles of Engineering Physics 2

  • Md Nazoor Khan, Simanchala Panigrahi(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  The point lattice may be regarded as the skeleton on which the actual crystal is built. 2 Principles of Engineering Physics 2 1.3 Fundamental Terms i. Point lattice The point lattice is defined as an array of points in space so arranged that every point has surroundings identical to that of every other point in the array. By identical surroundings we mean that, when we look in a particular direction putting ourselves at a lattice point, the same scenery is visible as that of any other point when we look in the same direction. A two-dimensional point lattice having infinite extension is shown in Fig. 1.1(a) and a three-dimensional point lattice assumed to have infinite extension is shown in Fig. 1.1(b). Point lattice, lattice or space lattice, are synonymously used. Figure 1.1 (a) A two-dimensional lattice. (b) A three-dimensional lattice. Observe that each point has identical surroundings. ABCD represents a unit cell selected in a two-different ways in a two-dimensional lattice and the heavily outlined one is the unit cell in a three-dimensional lattice ii. Unit cell As can be seen in Fig. 1.1(a), the entire two-dimensional lattice can be produced by translating the cell ABCD along the horizontal as well as vertical directions. Hence, ABCD is a unit cell. As has been illustrated in Fig. 1.1(b), the entire three-dimensional lattice can be produced by translating the heavily outlined cell in space in all possible directions. Therefore, the heavily outlined cell of Fig. 1.1(b) is a unit cell. Thus, the unit cell is defined as the smallest cell, translation of which generates the entire lattice. A three-dimensional general unit cell is shown in Fig. 1.1(b) as heavily outlined. iii. Crystallographic axes The vectors  1 a , 2 a  , and 3 a  that define the unit cell in Fig. 1.2 are called crystallographic axes of the unit cell. Thus, we can define the crystallographic axes of a unit cell as the three vectors defining the unit cell of a lattice.

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  Solid State Physics

  • Gerald Burns(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  SIMPLE CRYSTAL STRUCTURES Nothing happens in nature which can be attributed to the vice of nature, for she is always the same and everywhere one. Her virtue is the same, and her powers of acting; that is to say her laws and rules, according to which all things are and are changed from form to form, are everywhere and always the same; ... Spinoza, Ethics 3-1 Introduction In this chapter we discuss the basic, simple crystal structures. These structures can be, and often are, discussed without reference to any of the information that we acquired in the first two chapters. However, a discussion within the framework of this information gives a much deeper meaning to the structures and lays the groundwork for all crystal structures no matter how complicated. A crystal structure is a periodic arrangement of atoms. Focusing for a moment on a lattice point, there is some ordered arrangement of atoms about the lattice point (or lattice points in a multiply primitive unit cell). This arrangement repeats throughout all space by the translational symmetry of the lattice. This arrangement of atoms about a lattice point is called a basis or lattice complex. We can define a crystal structure by the particular lattice (which means the axial lengths, angles between them, and the centering, i.e., Ρ, I, F, C, or R) and the basis. Thus, we say crystal structure = lattice + basis (3-1) (This is not a mathematical equation, but just a concise way to write the above definition.) 3-2 Several Cubic Symmorphic Structures In this section we discuss a number of quite simple, cubic crystal structures. These structures are discussed in different subsections, listed by space group, in order to emphasize the variety of crystal structures that can have the same symmetry operations. 3-2a Space group Pm3m Figure 3-1 shows three different crystal structures, all with the same O h 1 ( P n i 3 m ) space group. The structure with atoms only at the lattice points of a cubic P-lattice (Fig.

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  Crystallography and Surface Structure

  An Introduction for Surface Scientists and Nanoscientists

  • Klaus Hermann(Author)
  • 2016(Publication Date)
  • Wiley-VCH

    (Publisher)

  As an example, the cubic unit cell of CsCl as well as the corresponding lattice vectors, shown in Figure 2.2, are primitive. On the other hand, replacing all cesium and chlorine atoms in Figure 2.2 by one atom type, for example, iron, yields a body-centered cubic (bcc) crystal. Here, the lattice vectors R 1 , R 2 , R 3 , shown in the figure, are non-primitive, since vector r 2 now becomes a lattice vector inside the morphological unit cell. In a crystal, the morphological unit cell contains, in general, p atoms at positions given by vectors r 1 , … , r p (lattice basis vectors), which form the basis of the crystal structure (the basis is sometimes also called the structure). Each atom at r i carries a label characterizing its properties, such as its nuclear charge or element name. These labels, usually omitted in the following, will be attached to each lattice basis vector if needed. For example, a definition r Cl 3 would refer to a chlorine atom 2.2 Representation of Bulk Crystals 11 placed at a position given by the third lattice basis vector. All lattice basis vectors r i inside the morphological unit cell can be written as linear combinations of the lattice vectors R 1 , R 2 , R 3 according to r i = x i R 1 + y i R 2 + z i R 3 , i = 1 … p (2.11) where x i , y i , and z i are real-valued coefficients with |x i | < 1, |y i | < 1, |z i | < 1. This use of relative coordinates x i , y i , z i to describe atoms inside the unit cell is com- mon practice in the crystallographic literature [28–33]. Note that, according to definition (2.11) the coefficients x i , y i , z i are generally not connected with the Cartesian coordinate system but with coordinate axes given by the lattice vectors R 1 , R 2 , R 3 . The origin of the morphological unit cell inside a crystal can always be chosen freely since the complete infinite crystal consists of a periodic arrangement of unit cells in three dimensions.

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  Kittel's Introduction to Solid State Physics

  • Charles Kittel(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  The diffraction work proved decisively that crystals are built of a periodic array of atoms or groups of atoms. With an established atomic model of a crys-tal, physicists could think much further, and the development of quantum the-ory was of great importance to the birth of solid state physics. Related studies have been extended to noncrystalline solids and to quantum fluids. The wider field is known as condensed matter physics and is one of the largest and most vigorous areas of physics. 3 Lattice Translation Vectors An ideal crystal is constructed by the infinite repetition of identical groups of atoms (Fig. 2). A group is called the basis . The set of mathematical points to which the basis is attached is called the lattice . The lattice in three dimensions may be defined by three translation vectors a 1 , a 2 , a 3 , such that the arrange-ment of atoms in the crystal looks the same when viewed from the point r as when viewed from every point r translated by an integral multiple of the a ’s: (1) Here u 1 , u 2 , u 3 are arbitrary integers. The set of points r defined by (1) for all u 1 , u 2 , u 3 defines the lattice. The lattice is said to be primitive if any two points from which the atomic arrangement looks the same always satisfy (1) with a suitable choice of the in-tegers u i . This statement defines the primitive translation vectors a i . There is no cell of smaller volume than a 1 a 2 a 3 that can serve as a building block for the crystal structure. We often use the primitive translation vectors to de-fine the crystal axes , which form three adjacent edges of the primitive paral-lelepiped. Nonprimitive axes are often used as crystal axes when they have a simple relation to the symmetry of the structure. 4 1 Crystal Structure Figure 2 The crystal structure is formed by the addition of the basis (b) to every lattice point of the space lattice (a). By looking at (c), one can recognize the basis and then one can abstract the space lattice.

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  Physical Chemistry

  • David Ball(Author)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 21.7 | Rationalizing Unit Cells 767 Unless otherwise noted, all art on this page is © Cengage Learning 2014. Despite an unstated presumption, unit cells are not invariant for a given com-pound. Different unit cells may be preferred under different conditions of tem-perature and pressure. These are examples of solid-solid phase changes. The easiest illustrations are for elemental materials. Perhaps one of the best-known differ-ences in unit cells is for elemental carbon, which has two common forms: graphite (a hexagonal unit cell, but not hcp) and diamond (face-centered cubic). Elemen-tal iron, for example, is body-centered cubic below about 910°C, but between 910° and about 1400°C it becomes face-centered cubic. Metallic tin is tetragonal at room temperatures, but below about 13°C (which is not much below room temperature!) it adopts a cubic structure. This causes a major problem because in doing so, the unit cell increases in volume by over 20%. Temperature-dependent solid-state phase changes are a major engineering concern. For molecular elements and compounds, the reasons for having a particular unit cell are complex and will not be considered. Generally, such materials adopt a unit cell that minimizes the overall energy of the compound. The choice of unit cell is therefore highly dependent on the molecule itself. There are also some marked solid-solid phase changes in molecular compounds. A well-known example is H 2 O. Many unit cells are actually known for solid H 2 O; that which we call “ice” is simply the stable crystalline phase at normal conditions of temperature and pressure.

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  Introduction to Physics

  Mechanics, Hydrodynamics Thermodynamics

  • P. Frauenfelder, P. Huber(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  We then have the three-layer or cubic closest packing of spheres. This type of structure is shown in four steps of construction in Fig. 9-19, Plate V. It can be seen that the face-centered cubic lattice, with a space diagonal of the cube verti-cal corresponds exactly to this type of packing since the triangles in Fig. 9-16 are arranged with their apices pointing alternately in opposite directions. 408 INTRODUCTION TO PHYSICS Element Al y-Fe (1100°C) jff -Ni Cu Ag Ir Pt Au Pb β (A) 4.049 3.64 3.524 3.615 4.086 3.839 3.924 4.078 4.950 Note that the basic cells of the primitive cubic, body-centered cubic, and face-centered cubic crystal structures are identical to the unit cells of the space lattices of the same name (Fig. 9-12). This is true in the other crystal systems also. (d) The diamond-type structure (A4 type, Fig. 9-20). Ζ = 8. FIG. 9-20. Basic cell of the diamond type; two interlocking face-centered cubic lattices, with a displacement of one-fourth the body diagonal, BD/4. Basic coordinates: 0 0 0 i i i f i * . This structure consists of two interpenetrating face-centered cubic lattices, one of which is displaced relative to the other along J of a body diagonal. Because of this displacement, the structure does not have the high degree of symmetry of the cubic system. There are three principal planes of symmetry. Each atom is surrounded by 4 others (all equidistant) arranged in a tetra-hedron (one of which is shown by dotted lines). Since the face-centered cube consists of four interpenetrating elementary cubes, we have here an interlacing of 8 primitive cubic lattices. The coordination number Κ = 4. Both types of closest packing occur frequently in nature. Other types are theoretically possible, but seldom occur. The following elements crystallize in a face-centered cubic lattice (room temperature unless otherwise given):

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