Hexagonal Close Packed

5 Key excerpts on "Hexagonal Close Packed"

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  Integrated Computational Materials Engineering (ICME) for Metals

  Concepts and Case Studies

  • Mark F. Horstemeyer(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  235 Section II Hexagonal Close Packed (HCP) Materials The next section includes Chapters 8–11 associated with just Hexagonal Close Packed (HCP) materials. Most of the chapters here focus on magnesium alloys addressing both horizontal and vertical upscaling and downscaling in the context of Integrated Computational Materials Engineering (ICME). The figure below illustrates the atom positions for a HCP metal. Schematic of a Hexagonal Close Packed (HCP) crystal. Integrated Computational Materials Engineering (ICME) for Metals: Concepts and Case Studies, First Edition. Edited by Mark F. Horstemeyer. © 2018 John Wiley & Sons, Inc. Published 2018 by John Wiley & Sons, Inc. 237 8 Electrons to Phases of Magnesium Bi-Cheng Zhou 1 , William Yi Wang 1 , Zi-Kui Liu 1 , and Raymundo Arroyave 2 1 Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA 2 Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA 8.1 Introduction This chapter presents the case study of bridging two length scales of electrons and phases in the context of Integrated Computational Materials Engineering (ICME). In Chapter 2, the bridge between electrons and atoms is discussed, but here, we ignore the atomistic level and bridge the information from electrons to phases using CALPHAD. This sort of bridge is similar to the next chapter, except for two things: Chapter 4 employs a phase field theory as the mesoscale modeling framework and not CALPHAD and the focus here in Chapter 3 is on magnesium alloys. Magnesium (Mg) has the second lowest density of all structural metals (1.74 g/cm 3 ) after beryllium, and Mg alloys have superior specific stiffness and strength in comparison with many structural materials (Stalmann et al., 2001; Kainer and Kaiser 2003).

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  Random Non-Random Periodic Faulting in Crystals

  • M. T. Sebastian, P. Krishna, M.T. Sebastian, P, Krishna(Authors)
  • 2017(Publication Date)
  • Taylor & Francis

    (Publisher)

  2

  Stacking Faults in Close-Packed Structures

  2.1 THE DESCRIPTION OF CLOSE-PACKED STRUCTURES

  T he crystal structure of several metals, alloys and inorganic compounds can be described in terms of the close-packing of equal spheres. When equal spheres are packed together to achieve maximum density, each sphere comes in contact with a maximum number of like spheres. Figure 2.1 shows the close-packing of spheres in two dimensions. Each sphere is in contact with six other spheres and the symmetry of this layer is 6 mm. Therefore it is called a hexagonal close-packed layer. Let this layer be designated as an A layer and let a hexagonal unit cell be chosen as shown in Figure 2.1 with unit cell vectors a and b lying in the layer. The next identical layer of spheres can be close-packed on top of this layer in two ways. Either its spheres can occupy the sites marked B which lie in the voids oriented as Δ or they can occupy the sites marked C which lie in the voids oriented as ∇. Similarly, the third layer can be close-packed on top of the second one in two different ways. It is easy to see that there are only three possible positions for a close-packed layer in a three-dimensional packing – either A, B or C, corresponding to atomic positions 0, 0, Z; 1/2, 2/3, Z and 2/3, 1/3, Z in the hexagonal unit cell. The A, B and C layers are identical to each other but displaced. The B and C positions are obtained from the A position by displacing the layer through a vector ±S where S = a/3 〈101̅0〉.

  FIGURE 2.1

  Close-packing of spheres in two dimensions. The hexagonal unit cell is outlined. Co-ordinates of A, B and C sites are 0,0; 1/3, 2/3 and 2/3, 1/3 in the unit cell.

  Any sequence of the letters A, B and C represents a possible manner of close-packing equal spheres so long as no two successive letters are alike. In the resulting arrangement each sphere is in contact with 12 nearest neighbours −6 in its own layer, 3 in the layer above it and 3 in the layer below. This is the maximum number of spheres that can be arranged to touch a given sphere and it provides the maximum packing density of 0.7405 for an infinite lattice arrangement (1 ). There are, however, other arrangements of a finite number of equal spheres which have a higher packing density (2 ). If h denotes the perpendicular spacing between successive layers in an ideal close-packing, it is easy to calculate that the ratio or 0.8165. In actual crystal structures this is seldom achieved and deviations from this ideal value (3

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  Advances in Molecular Structure Research

  • Magdolna Hargittai, Istvan Hargittai(Authors)
  • 1995(Publication Date)
  • JAI Press

    (Publisher)

  Close packing can exist if the molecular coordination number is sufficiently high (usually 12). The real number depends on the specific shape of the molecule. By an array of Hexagonal Close Packed spheres stretching along an arbitrary direction, i.e., transforming them into triaxial ellipsoids, it is possible to obtain a model of packing of arbitrarily shaped bodies in its first approximation. Clearly, in this case too, layers of molecules, similar to those derived from the packing of spheres, can be identified; the coordination number of each molecule will be 12, with a distribution of neighbors identical to that existing in the packing of spherical molecules. The geometrical model is fundamental to our understanding of the packing of molecular crystals, as it is the close packing model in metals [9,10] and in ionic structures [11]. Recently, Frank [12] reviewed the definition of this concept for nonrigid spheres in a metal. He shows that the best occupancy of space is given, in this case, by body centered cubic (bcc) structures relative to the cubic close packed (ccp) structures. In fact, at equal density, in bcc structures the nuclei are closer to each other, thus better filling the available space. This can be the reason why many metals attain and retain the bcc structure at very high pressure. Here emphasis is given to distances, neglecting the differences in the coordination number (CN) which is a factor that must be taken into consideration when dealing with stability. It is feasible that this observation could be extended to organic molecular structures, when atoms are not considered uncompressible. It is possible that a structure that cannot be described in the classical close-packing model is, in fact, close packed in this new approach. Both coordination number and distances have been considered by Pearson [13] in the study of the cohesive energies of simple AB ionic and covalent solids.

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  Introduction to Condensed Matter Physics

  Volume 1

  • Feng Duan, Jin Guojun;;;(Authors)
  • 2005(Publication Date)
  • WSPC

    (Publisher)

  In this case, the formation of a definite number of bonds or links between neighboring disks will play the dominant role in structure-building (see Table 2.2.1). Table 2.2.1 Coordination number, packing and covering density for some 2D structure. Structure type Coordination number Packing Covering density density hcp 6 0.9069. . . 1.2092. . . square 4 0.7854. . . 1.5708. . . honeycomb 3 0.6046. . . It should be noted that in 2D structure there is an outstanding structure, i.e., hexagonal close-packing, in which the highest packing density, highest coordination number and least covering density are simultaneously achieved. Kepler (1611) in his booklet “On Hexagonal Snow” gave the first scientific conjecture about closest sphere packings in 2D and 3D space. For 3D packing, first hexagonal close-packed layers of spheres are formed, then the upper layer stacks on the voids of the lower layer, and so ad infinitum . Two main types of close-packed structures are formed: (a) abcabc . . . → fcc, (b) ababab . . . → hcp. (a) (c) (b) Figure 2.2.3 Unit cells of three kinds of packings. (a) fcc; (b) bcc; (c) hcp. (a) (b) (c) Figure 2.2.4 Wigner–Seitz cells of three kinds of packings. (a) fcc; (b) bcc; (c) hcp. In the fcc (face-centered-cubic) structure, every sphere is situated on a lattice point, i.e., lattice packing; while in the hexagonal close-packed (hcp) structure, only one half of the spheres are situated on lattice points; besides hcp structure there are an infinite set of stacking sequences to fulfill the closed-packed condition. All these closed-packed structures have identical packing densities f p = 0 . 74048 . . ., and the coordination number for closed packed structures is z = 12. The WS cell for the fcc structure is the rhombic dodecahedron, and the reciprocal lattice of fcc is bcc, whose WS cell is the truncated octahedron. Though bcc has lower packing density, it has thinner covering

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  Advanced Electrical and Electronics Materials

  Processes and Applications

  • K. M. Gupta, Nishu Gupta, Ashutosh Tiwari(Authors)
  • 2015(Publication Date)
  • Wiley-Scrivener

    (Publisher)

  As already discussed, a crystalline form of solid has periodically repeated arrangement of atoms. But the solids in non-crystalline form do not have long range periodic repetition. However, in both of the above forms, the coordination number is almost the same. Formation of non-crystalline structure is characterized by several factors enumerated below:

  1. Non-formation of three-dimensional primary bond.
  2. Formation of one-dimensional chain molecule.
  3. Formation of two-dimensional sheet molecule.
  4. Absence of primary bonds in all the directions.
  5. Weak secondary bond.
  6. Non-parallel, entangled chain configuration.
  7. Open network of the atomic packing.

  Figure 3.1 shows the difference in structures of crystalline and non-crystalline solids.

  Figure 3.1

  Structure of (a) crystalline solid, and (b) non-crystalline solid.

  3.3 Hexagonally Closed Packed Structure (HCP)

  This is also known as closed packed hexagonal (CPH) structure. Arrangement of atomic packing and sequential piling of atoms in HCP is of …ABAB… form as shown in Fig. 3.2 . We will study more about the geometry of HCP crystal now.

  Figure 3.2

  A hexagonally closed packed (HCP) structure.

  HCP structure is different from hexagonal structure. It is more denser than the hexagonal one. A HCP unit cell is shown in Fig. 3.2 . There are total seventeen atoms in it. Six atoms are placed on each of the bottom and top hexagonal corners, one each on bottom and top hexagonal faces, and three atoms on vertical alternate planes. The atoms are closed packed, but are shown separated for clarity. The atoms in planes A are placed at lattice points but the atoms in plane B are not.

  The geometrical details of HCP unit cell are given as follows:

  Effective number of lattice points	3
  Effective number of atom	6
  Basis (effective number of atoms/effective number of lattice points)	6/3=2
  Coordination number	12
  c/a ratio for a perfect crystal	1.633
  Atomic packing factor	0.74

  3.4 VOIDS

  The closed packing of atoms are redrawn in Fig. 3.3 showing the plan (top view). The space enclosed by atoms marked V are vacant. The vacant space between atoms is called void. These voids are also known as interstitial voids

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