Physics
Body Centered Cubic
Body Centered Cubic (BCC) is a crystal structure in which atoms are arranged in a cubic lattice with one atom at the center of the cube and one at each corner. This structure is characterized by a coordination number of 8 and is commonly found in metals such as iron and chromium. BCC crystals have a higher packing density than simple cubic structures.
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eBook - PDF- Sharon Ann Holgate(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
a Cube diagonal O N P a FIGURE 2.19 In an elemental bcc crystal, the atoms “touch” along the [111] cube diagonal. (The notation for directions and planes in unit cells is discussed in Section 2.2.7.) Crystal Clear 35 Although atoms are not, of course, hard spheres touching each other, the rough estimates of atomic density and radius that equations like these give are useful when calculations need to be done on properties that are affected by structure. As we will see in Chapter 4, ductility and brittleness are among the mechanical properties affected by crystal structure. Also, as Box 4.1 reveals, strain is inherent in quantum dots because of differences in either interatomic spacing or atomic sizes between the two different types of semiconductor that the dots are made from. The distances between the atoms in solids (and therefore an estimate of the atomic radius) can be obtained experimentally via X-ray diffraction, which will be discussed in detail in Chapter 5. Face-Centred Cubic Structure Another way for atoms to be closely packed together is to have the face-centred cubic (fcc) structure shown in Figure 2.20(a). Instead of having an extra lattice point at the centre of each cube like the bcc structure, there is a lattice point at the centre of each face. So the fcc lattice looks rather like a die with a 1 on every face. (“Die” is the singular of the word “dice.”) The metals copper (Cu), silver (Ag), and gold (Au) that make up Group 1B of the periodic table have the fcc structure, as do several transi-tion metals, including nickel (Ni) and platinum (Pt). Other elements crystallising in the fcc structure include the metals aluminium (Al), strontium (Sr), and calcium (Ca) and the van der Waals-bonded solids of neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn). An fcc unit cell contains a total of four lattice points, and so elements with this structure (that have a basis of one) have four atoms per unit cell.
- Donald Askeland, Wendelin Wright(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
BCC cells have a packing factor of 0.68, and SC cells have a packing factor of 0.52. Notice that the packing factor is independent of the radius of atoms, as long as we assume that all atoms have a fixed radius. What this means is that it does not matter whether we are packing atoms in unit cells or packing basketballs or ping-pong balls in a cubical box. The maximum achievable packing factor is H9266 / Ï 18 ! This discrete geometry concept is known as Kepler’s conjecture . Johannes Kepler proposed this conjecture in the year 1611, and it remained an unproven conjecture until 1998 when Thomas C. Hales actually proved this to be true. Example 3-5 Determining the Density of BCC Iron Determine the density of BCC iron, which has a lattice parameter of 0.2866 nm. SOLUTION There are two atoms per unit cell in BCC iron. Volume of unit cell 5 s 2.866 3 10 2 8 cm d 3 5 2.354 3 10 2 23 cm 3 Density H9267 5 s number of atoms/cell ds atomic mass d s volume of unit cell ds Avogadro number d H9267 5 s 2 atoms/cell ds 55.847 g/mol d s 2.354 3 10 2 23 cm 3 /cell ds 6.022 3 10 23 atoms/mol d 5 7.879 g/cm 3 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 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. 65 3-3 Lattice, Basis, Unit Cells, and Crystal Structures The Hexagonal Close-Packed Structure The hexagonal close-packed structure (HCP) is shown in Figure 3-11. The lattice is hexagonal with a basis of two atoms of the same type: one located at (0, 0, 0) and one located at (2/3, 1/3, 1/2).
eBook - PDFMetals and Materials
Science, Processes, Applications
- R. E. Smallman, R J Bishop(Authors)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
Commencing with an atom (or group of Atomic arrangements in materials 19 Position of the centre of the atom (a) (b) (c) Figure 2.11 Arrangement of atoms in (a) face-centred cubic structure, (b) close-packed hexagonal structure, and (c) body-centred cubic structure. atoms) at either a lattice point or at a small group of lattice points, a certain combination of symmetry operations will ultimately lead to the three-dimen-sional development of any type of crystal structure. The procedure provides a unique identifying code for a structure and makes it possible to locate it among 32 point groups and 230 space groups of symmetry. This classification obviously embraces the seven crystal systems. Although many metallic structures can be defined relatively simply in terms of space lattice and one or more lattice constants, complex structures require the key of symmetry theory. 2.5 Selected crystal structures 2.5.1 Pure metals We now examine the crystal structures of various elements (metallic and non-metallic) and compounds, using examples to illustrate important structure-building principles and structure/property relations. 1 Most elements in the Periodic Table are metallic in character; accordingly, we commence with them. Metal ions are relatively small, with diameters in the order of 0.25 nm. A millimetre cube of metal therefore contains about 10 20 atoms. The like ions in pure solid metal are packed together in a highly regular manner and, in the majority of metals, are 'Where possible, compound structures of engineering importance have been selected as illustrative examples. Prototype structures, such as NaCl, ZnS, CaF 2 , etc., which appear in standard treatments elsewhere, are indicated as appropriate. packed so that ions collectively occupy the minimum volume.
eBook - PDF- Nevill Gonzalez Szwacki, Teresa Szwacka(Authors)
- 2016(Publication Date)
- Jenny Stanford Publishing(Publisher)
Therefore, the bcc and fcc lattices may be described in terms of cubic, rhombohedral, and hexagonal axes by using cubic (body centered or face centered), primitive rhombohedral, and triple hexagonal R unit cells, respectively. In Fig. 2.41 we show the three types of cells of the bcc lattice by putting them all together and with a common origin. Some information about those cells is listed in Table 2.6 and in Table 2.7 is listed the same information, but for the three types of unit cells of the fcc lattice. Triple Hexagonal R Cell in the Cubic Lattice 83 Figure 2.41 Three types of unit cells of the bcc lattice. Each of the three triple hexagonal R cells shown in the figure is defined by basis vectors a h , b h , and c h or their linear combinations. Inside the hexagonal prism there is a rhombohedral P unit cell defined by basis vectors a r , b r , and c r . Besides that, there is a cubic I cell of the bcc lattice defined by vectors a c , b c , and c c . All three unit cells have the same origin O . 84 Three-Dimensional Crystal Lattices Table 2.6 Basic information about three types of unit cells of the bcc lattice Body Centered Cubic Lattice Unit cell type Cell parameters Number of lattice points per cell Cubic I a c 2 Rhombohedral P a r = ( √ 3 / 2) a c α r = 109 ◦ 28 1 Triple hexagonal R a h = √ 2 a c c h = ( √ 3 / 2) a c 3 Table 2.7 Basic information about three types of unit cells of the fcc lattice Face Centered Cubic Lattice Unit cell type Cell parameters Number of lattice points per cell Cubic F a c 4 Rhombohedral P a r = ( √ 2 / 2) a c α r = 60 ◦ 1 Triple hexagonal R a h = ( √ 2 / 2) a c c h = √ 3 a c 3 2.14 Wigner–Seitz Cell 2.14.1 Introduction The primitive unit cells that we have considered, until now, for the case of centered Bravais lattices, do not have the point symmetry of the lattice. However, each Bravais lattice has a primitive unit cell that has the point symmetry of the lattice.
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