Physics
Cubic Close Packing
Cubic close packing is a method of arranging atoms or spheres in a crystal structure. It involves stacking layers of spheres in a way that maximizes the amount of space filled while minimizing the empty spaces between them. This arrangement is commonly found in metals and other materials.
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6 Key excerpts on "Cubic Close Packing"
eBook - ePub- M. T. Sebastian, P. Krishna, M.T. Sebastian, P, Krishna(Authors)
- 2017(Publication Date)
- Taylor & Francis(Publisher)
Since three-dimensional close-packings of spheres are built by stacking identical close-packed layers of the kind shown in Figure 2.1, they all have the same a and b dimensions of the unit cell while the c dimension depends on the number of layers after which the A, B, C sequence of layers repeats itself. Theoretically, there are n layers to be stacked over each other. There are 2 n−1 ways in which they can be close-packed since each layer can be placed over the previous one in two different ways. Of all these the ones that most commonly occur among metals and inorganic compounds are those which correspond to the ABAB … or hexagonal close-packing (hcp) and the ABCABC … or the cubic close-packing (ccp). However, there are certain special materials like SiC, ZnS, PbI 2, CdI 2, etc, which are known (4) to crystallise in a large variety of close-packed structures called poly types, which have larger repeat periods along the c-direction. In these materials one also encounters a number of different close-packed structures having the same identity period of n layers but with the layers stacked differently in the unit cell. Before fully describing such structures it is necessary to understand the type of voids that occur in a close-packed structure. 2.2 THE VOIDS IN A CLOSE-PACKING All three-dimensional close-packings of equal spheres have two kinds of voids in them. When the triangular void Δ or ∇ in a close-packed layer has a sphere placed directly above it, there results a tetrahedral void with four spheres round it, as shown in Figure 2.2(a). Since each sphere in the close-packing has 3 such spheres in the layer above it and 3 in the layer below it, there are two tetrahedral voids per sphere in an infinite close-packing. If R denotes the radius of each sphere in the close-packing it is easy to show that the largest sphere that can fit into the tetrahedral void has a radius given by r = 0.225R. The other kind of void in a close-packing is called an octahedral void
eBook - PDF- Richard Zallen(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
This is an admirable and time-honored tradition, and has yielded (and will continue to yield) valuable insights. However, it is also true that it is frequently perilous to attempt to generalize specific results and conclusions from one dimensionality to another. We shall encounter several illustrations of this caveat throughout this book. In fact, we have already done so during the course of our discussion of random close packing, and will now make this ex- plicit. One may reasonably adopt the viewpoint that the geometric stability and consequent physical significance of the rcp structure arises from the following circumstance: In remarkable contrast to one and two dimensions, shoa-range dense packing and long-range crystalline order are not concordant with each other in three dimensions. The construct of an rcp structure is not meaningful in one dimension. A linear array of movable nonoverlappable line segments of equal length (the Id equivalent of identical hard spheres) necessarily collapses under compression to a crystal lattice. Likewise, experience also indicates the absence of a stable rcp structure in two dimensions. An agitated and compressed planar array of equal coins tends to aggregate in domains of 2d crystalline close packing. In both one and two dimensions, there exists an intrinsic consistency between the crystalline close-packed (ccp) structure and the demands of the most efficient 2.4 RANDOM CLOSE PACKING 59 local packing. But in three dimensions this is not true. These assertions are eas- ily demonstrated as follows. The largest number of d-dimensional spheres which can be locally packed so that each contacts every one of the others is 2 ford = 1, 3 ford = 2, 4 ford = 3 (and d + 1 ford = d, an example of an in- terdimensionality generalization which works). The resulting d-dimensional polyhedral hole formed by taking the sphere centers as vertices is a line seg- ment, equilateral triangle, and regular tetrahedron for d = 1, 2, and 3, respectively.
- Donald Askeland, Wendelin Wright, Donald Askeland(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
This structure is known as the hexagonal close-packed structure (HCP). Metals with only metallic bond- ing are packed as efficiently as possible. Metals with mixed bonding, such as iron, may have unit cells with less than the maximum packing factor. No commonly encountered engineering metals or alloys have the SC structure. Density The theoretical density of a material can be calculated using the properties of the crystal structure. The general formula is Density r 5 snumber of atoms/celldsatomic massd svolume of unit celldsAvogadro numberd (3-5) If a material consists of different types of atoms or ions, this formula has to be modified to reflect these differences. Example 3-5 illustrates how to determine the density of BCC iron. Since for FCC unit cells, a 0 5 4r/Ï2 : Packing factor 5 s4d 1 4 3 pr 3 2 s4ryÏ2 d 3 5 p Ï18 > 0.74 The packing factor of p/Ï18 > 0.74 in the FCC unit cell is the most efficient packing possible. 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 p/Ï18 ! 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.
eBook - PDF- 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
eBook - PDF- Eugene A. Irene(Author)
- 2005(Publication Date)
- Wiley-Interscience(Publisher)
Now to form the third layer, there are two possibilities. If the possibility that the third layer forms in direct correspondence to the A layer, then this third layer is also named an A layer (another A layer). The close packing of layers follows the order A B A B A. . . . This form has hexagonal symmetry and is 2 3 2.6 LATTICE GEOMETRY 23 b) a) c) a = 2R o a o 4R 3 a = o 4R 2 a = o } Figure 2.11 Close packing directions for (a) PC, (b) BCC, and (c) FCC cubic unit cells where the closely packed direction is indicated by the touching of atoms (shaded ). The relationship between the lattice para- meter a 0 and the atomic radius R is also given. consequently called hexagonal close packed (HCP). Alternatively, if the third layer forms in the other position, which registers neither with the A or B layers, it forms a C layer with the order A B C A B C. . . . This packing is also close packing and possesses FCC symmetry, so it is termed accordingly. Atoms of both HCP and FCC close packing are shown in Figure 2.12. 2.7 THE WIGNER-SEITZ CELL Up until now we have chosen the unit cell boundaries somewhat intuitively by extract- ing a portion of the larger lattice. It is reasonable to expect that by this method the care- fully chosen pieces will reproduce the entire lattice by translation and therefore fulfill the unit cell definition. There are other methods to select the unit cell that keep the require- ments the same, namely that the unit cell must contain the symmetry of the lattice and fill all space by translation. It is particularly useful in some applications to choose a cell that is primitive, a cell that contains a single lattice point. One way to do this is with a square 2-D lattice as depicted in Figure 2.13. As the figure shows, one starts at any lattice point in the 2-D array and draws lattice vectors emanating from the starting point to first nearest neighbors (solid arrows). The bisectors (dashed lines) of these vectors are con- structed and extended.
eBook - PDF- Magdolna Hargittai, Istvan Hargittai(Authors)
- 1995(Publication Date)
- JAI Press(Publisher)
328 LAURA E. DEPERO either through a juxtaposition of regularly displaced layers for which the repeat vector forms an arbitrary angle with the layer plane, or by inversion centers, glide planes, and screw axes. For molecules without symmetry elements, close packing is attainable in the space groups P1, P2 l, P21/c, Pbca, Pna2 l, and P212121. For centrosymmetric molecules, the number of groups in which close packing can be achieved is even smaller: they are P1, P21/c, C2/c, and Pbca. In a recent paper, Wilson [136] discusses the Kitaigorodsky categorization of space groups, in which the space groups are divided into four categories: (1) closely-packed (PI, P21, P21/c, C2/c), (2) limiting close-packed (C2/m), (3) permissible (P1, C2, Cc, Cm, P21/m), and (4) impossible (P23, Pm, P2/m). This categorization has generally proved very successful, with only a few exceptions. Indeed, as a consequence of a particular molecular shape, some impossible space groups become possible; in some space groups, requiting the molecules to capital- ize on point-group symmetry, the molecules are in general positions (Figure 7). If the bonding forces are directional, as in a covalent crystal, the energy minimum is attained upon the saturation of the directional covalent bonds. In these structures, atoms have relatively few neighbors, as, for example, in diamond and NiAs. The symmetry of these structures is defined by the coordination symmetry of the constituent atoms. It is interesting to note that the volume per atom in structures with different types of bonds differs only slightly, on the average, since the covalent bonds are usually slightly shorter than the ionic or metallic ones. B. SpaceGroup Statistics In the cases of close packing, the coordination number of 6 in a layer can be achieved with any mutual orientation of the molecules. The space group P21/c, in which both centrosymmetric and non-centrosymmetric molecules can be packed, is the most frequent.
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