The tetrahedron is just an imaginary construction whose four corners point to the centers of the four atoms that are in contact. Octahedral sites are larger than tetrahedral sites. An octahedron has six corners and eight sides. We usually draw octahedra as a double square pyramid standing on one corner left , but in order to visualize the octahedral shape in a close-packed lattice, it is better to think of the octahedron as lying on one of its faces right.

Each sphere in a close-packed lattice is associated with one octahedral site, whereas there are only half as many tetrahedral sites. This can be seen in this diagram that shows the central atom in the B layer in alignment with the hollows in the C and A layers above and below. The face-centered cubic unit cell contains a single octahedral hole within itself, but octahedral holes shared with adjacent cells exist at the centers of each edge. Added to the single hole contained in the middle of the cell, this makes a total of 4 octahedral sites per unit cell.

### Close-Packing of Identical Spheres

This is the same as the number we calculated above for the number of atoms in the cell. Many pure metals and compounds form face-centered cubic cubic close- packed structures. The existence of tetrahedral and octahedral holes in these lattices presents an opportunity for "foreign" atoms to occupy some or all of these interstitial sites. In order to retain close-packing, the interstitial atoms must be small enough to fit into these holes without disrupting the host CCP lattice.

When these atoms are too large, which is commonly the case in ionic compounds, the atoms in the interstitial sites will push the host atoms apart so that the face-centered cubic lattice is somewhat opened up and loses its close-packing character. Alkali halides that crystallize with the "rock-salt" structure exemplified by sodium chloride can be regarded either as a FCC structure of one kind of ion in which the octahedral holes are occupied by ions of opposite charge, or as two interpenetrating FCC lattices made up of the two kinds of ions. The two shaded octahedra illustrate the identical coordination of the two kinds of ions; each atom or ion of a given kind is surrounded by six of the opposite kind, resulting in a coordination expressed as How many NaCl units are contained in the unit cell?

If we ignore the atoms that were placed outside the cell in order to construct the octahedra, you should be able to count fourteen "orange" atoms and thirteen "blue" ones. But many of these are shared with adjacent unit cells. Similarly, the center of an edge is common to four other cells, and an atom centered in a face is shared with two cells. Taking all this into consideration, you should be able to confirm the following tally showing that there are four AB units in a unit cell of this kind. The space-filling model on the right depicts a face-centered cubic unit cell of chloride ions purple , with the sodium ions green occupying the octahedral sites.

Since there are two tetrahedral sites for every atom in a close-packed lattice, we can have binary compounds of or stoichiometry depending on whether half or all of the tetrahedral holes are occupied. Zinc-blende is the mineralogical name for zinc sulfide, ZnS. An impure form known as sphalerite is the major ore from which zinc is obtained. This structure consists essentially of a FCC CCP lattice of sulfur atoms orange equivalent to the lattice of chloride ions in NaCl in which zinc ions green occupy half of the tetrahedral sites.

## Evolution of the dense packings of spherotetrahedral particles: from ideal tetrahedra to spheres

As with any FCC lattice, there are four atoms of sulfur per unit cell, and the the four zinc atoms are totally contained in the unit cell. Each atom in this structure has four nearest neighbors, and is thus tetrahedrally coordinated. It is interesting to note that if all the atoms are replaced with carbon, this would correspond to the diamond structure.

Fluorite, CaF 2 , having twice as many ions of fluoride as of calcium, makes use of all eight tetrahedral holes in the CPP lattice of calcium ions orange depicted here. To help you understand this structure, we have shown some of the octahedral sites in the next cell on the right; you can see that the calcium ion at A is surrounded by eight fluoride ions, and this is of course the case for all of the calcium sites. Since each fluoride ion has four nearest-neighbor calcium ions, the coordination in this structure is described as In Section 4 we saw that the only cubic lattice that can allow close packing is the face-centered cubic structure.

The simplest of the three cubic lattice types, the simple cubic lattice , lacks the hexagonally-arranged layers that are required for close packing. But as shown in this exploded view, the void space between the two square-packed layers of this cell constitutes an octahedral hole that can accommodate another atom, yielding a packing arrangement that in favorable cases can approximate true close-packing. Each second-layer B atom blue resides within the unit cell defined the A layers above and below it.

The A and B atoms can be of the same kind or they can be different. If they are the same, we have a body-centered cubic lattice. If they are different, and especially if they are oppositely-charged ions as in the CsCl structure , there are size restrictions: if the B atom is too large to fit into the interstitial space, or if it is so small that the A layers which all carry the same electric charge come into contact without sufficient A-B coulombic attractions, this structural arrangement may not be stable.

CsCl is the common model for the BCC structure. As with so many other structures involving two different atoms or ions, we can regard the same basic structure in different ways. Thus if we look beyond a single unit cell, we see that CsCl can be represented as two interpenetrating simple cubic lattices in which each atom occupies an octahedral hole within the cubes of the other lattice.

Chem1 Virtual Textbook. Learning Objectives Make sure you thoroughly understand the following essential ideas: The difference between square and hexagonal packing in two dimensions. The definition and significance of the unit cell.

Sketch the three Bravais lattices of the cubic system, and calculate the number of atoms contained in each of these unit cells. Show how alternative ways of stacking three close-packed layers can lead to the hexagonal or cubic close packed structures. Explain the origin and significance of octahedral and tetrahedral holes in stacked close-packed layers, and show how they can arise. Close-Packing of Identical Spheres Crystals are of course three-dimensional objects, but we will begin by exploring the properties of arrays in two-dimensional space. This will make it easier to develop some of the basic ideas without the added complication of getting you to visualize in 3-D — something that often requires a bit of practice.

Suppose you have a dozen or so marbles. How can you arrange them in a single compact layer on a table top? Obviously, they must be in contact with each other in order to minimize the area they cover. The packing of constituent particles inside lattice in such a way that they occupy maximum available space in the lattice is known as Close Packing. In one dimension close packing, the spheres are arranged in a row touching each other. In one-dimensional close packing, each sphere is in direct contact with two of its neighbor spheres. The number of nearest spheres to a particle in a lattice is called Coordination Number.

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Since there are two spheres in direct contact with the one sphere, the coordination number of one-dimensional close packing is 2. Two-dimensional close packing is done by stacking rows of hard spheres one above the other. This can be done in two ways:. The packing in which one sphere touches two spheres placed in two different rows one above and one below is called AAA type close packing. The coordination number of AAA type two-dimensional close packing is 4. The AAA type is formed by placing one-dimensional row directly one above the other in both horizontal and vertical directions.

It is also called two-dimensional square close packing as the rows of spheres when arranged in vertical and horizontal alignments form a square. The packing in which the spheres in the second row are located in the depressions of the first row. The coordination number of ABA Type packing is 6 as each sphere is in direct contact with 6 other spheres. The formation of real lattices and structures take place through three-dimensional close packing.

They are formed by stacking two-dimensional layers of spheres one above the other. This can be done by two ways:. Formation of three-dimensional close packing can be done by placing the second square closed packing exactly above the first one. In this close packing, the spheres are aligned properly in horizontally and vertically. Similarly, by placing more layers one above the other, we can obtain a simple cubic lattice.

The unit cell of the simple cubic lattice is called the primitive cubic unit cell. Models could not only give an intuitive image of structure, but also provide a foundation for developing high performance of new glassy materials. The atomic structure determines the intrinsic properties of metallic glasses. We summarized the atomic structural models up to date and show them as follows. Compared with the structures of gas and crystalline solid, the structure model of liquid was mysterious for a long time. In, Bernal 47 first put forward a model to describe the structure of liquid called random dense packing model.

This model mainly contains five Bernal polyhedrons 48 with edges of equal length, as shown in Figure 1 , and their percentages are listed in Table 1. In , he 49 further pointed out that a random dense packing of hard spheres resembles a monatomic liquid quite closely. In this model, the main assumptions are as follows: 1 the liquid is homogenous, coherent and irregular; 2 all the atoms are regarded as rigid balls and stacking without rules, and there is no extra hole whose size is an atom volume; 3 the atoms are incompressible.

In , Cohen and Turnbull 50 found that Bernal dense random model can describe the hypothetical metastable glassy state of simple liquids. Finney used Voronoi tessellation 51 , 52 to study the random dense packing model. He analyzed the Voronoi polyhedrons and found that the average face number of Voronoi polyhedra is This shows that there are a large number of pentagons in voronoi polyhedra. Five-rotating symmetry could not form the long-range order and hence the structures of Figures 1 c , d and e are possible to form the amorphous alloy.

In , Cargill 53 compared the experimental pair distribution function PDF for noncrystalline Ni-P alloys with the distribution function for random dense packing of hard spheres and found that the feature was greatly similar. Besides, he pointed out that the random dense packing models may be useful in interpreting properties of amorphous alloys. The PDF 54 is often used to describe and distinguish amorphous structure which is related to the probability of finding the center of a particle at a given distance from the center of another particle. For short distance, the PDF is related to the stack structure of particles.

However, the curve of the PDF gradually converges to unity at a larger distance, showing the long-range disorder. The partial PDF is given by:. Figure 2 is the schematic illustration of PDF.

The second, third, fourth, and fifth peaks are blunt but more pronounced than those found in liquids Based on the random packing of hard spheres model, Polk's simulation results agreed well with the RDF obtained by experiments on metal-metalloid metallic glasses This model could explain why the metallic glasses cannot exhibit the long-range order structure in a certain extent and the RDF agrees well with the experimental results especially on metal-metalloid metallic glasses. The RDF 58 , 59 of the model was simulated and compared with the experimental results of Ni-P glassy alloys by Bennett et al.

They found that the first peak suits well with the experimental result, but the second peak has certain differences. Sadoc et al. Their calculated values of some metallic glassy systems meet well with the experimental data, but the density is lower than the dense packing. In order to solve the contradiction between local structure of second peak and density, Connell 61 , Baker et al.

The modified model showed more agreement with the actual materials compared with the Bernal model. The spheres are softened when considering the potential energy. According to the modern physics, we know that the atomic radius will change in the process of atomic interaction. So it is an important progress from hard spheres to soft spheres concept.

Simplifying the system and problems, mainly focusing on the geometry of the atomic arrangement, provides a very fundamental knowledge and is a very good start for studying atomic structures. The random dense packing model has reached certain achievements on the analysis of some metallic glassy systems, such as alloys with constituent species having comparable atomic sizes and insignificant chemical short-range order, contributing to understanding amorphous alloys at the atomic level, even people have been struggling to describe many binary metallic glasses, especially metal-metalloid alloys.

In , Bragg 65 provided a dynamical model to describe the structure of crystal. He thought that the basic structure of metal consisted of small grains and grain boundaries. The model is called as micro-crystallite model. This model was initially used to describe the structure of liquid, since the structure of ambiguity is similar to the liquid-like structure. Afterwards, Bagley et al. A schematic illustration 67 of the model is given in Figure 3 , which can explain some XRD data of certain metallic glasses.

Inside of small grain, it exhibits the short-range order, which is similar to the crystal.

## Close-packing of equal spheres

These small grains are mixed randomly and their orientations are in disorder. It is therefore difficult to form the long-range ordered structure. The model can qualitatively explain some properties of amorphous alloys. However, the RDF is not agreement well with the experimental data. The details of the structures of microcrystalline area and boundary area are unknown, and furthermore the scientists also did not observe directly these areas in their experiments 68 , When the size of micro-crystallite is smaller, the volume of boundary will occupy too much and it is difficult to determine the atomic arrangements in associated areas.

It is therefore regarded that this model does not fit the structures of metallic glasses. Free volume model was put forward by Cohen and Turnbull 70 - 72 to explain the phenomenon of self-diffusion in Van der Waals liquids and liquid metals. Then Spaepen 73 developed this model by analysis of the mechanism for deformation and dynamics of metal in , which can be used to describe the deformation and fracture of metallic glasses. When the liquid is cooled slowly, the atoms will rearrange and form the crystalline solid.

In this process, the volume of material will shrink. On the other hand, when the liquid is quenched fast, the "gap" will be kept in amorphous solid. Its volume is larger than the corresponding crystalline solid and hence the increased part is called the free volume. The term of the free volume is a very important concept and plays a significant role in amorphous alloys, and has been widely used to explain physical and mechanical properties of amorphous alloys.

Figure 4. Figure 4 b shows the creation of the free volume. Under the applied stress, the certain amount of free volume is created when an atom squeezes into a smaller space whose volume is V. This model can analyze the flow behavior of metal glasses at high temperature. At low temperature, however, it cannot be well used to qualitatively analyze the deformation behavior of metallic glasses.

The free volume model provides a simple and practical model to describe the plasticity of amorphous alloys. The applicability of the model has been well verified and it can qualitatively explain many mechanics properties of amorphous alloys 74 - In , Argon 77 used "shear transformation" to explain the plastic deformation of metallic glasses and the theory meets well with the experimental observations.

He thought that the "flow event" was not a single isolated event, but a rearrangement of a group of atoms. Falk 78 , 79 developed the theory of "shear transformation zone STZ " according to the molecular dynamics simulation in conjunction with the complete mathematical model.

An STZ only appears in response to external stimuli, being undefinable a priori in the static glass structure before deformation The shear transformation zone theory is an extension and expansion for free volume model at the molecular level. It can solve the defects of the free volume model, particularly in the description of low temperature state. But this process involves too many parameters. Based on the researches of Stillinger et al. They well explained the plasticity behavior in low temperature.

Bernal's model has been widely accepted for metallic glasses, but it fails to describe the metal-metalloid-based alloys with pronounced chemical short-range order In light of this, Gaskell 87 - 89 put forward a model shown in Figure 5. He 87 thought that there was a basic structure unit that is similar to the corresponding component crystal microstructure in metallic glasses, this structure unit was regarded as tri-capped trigonal prism TTP that can form the net by coplanar points and coplanar faces.

Figure 5 a displays the regular trigonal prismatic coordination, in which N is the non-metallic element, M I are metal atoms, M II are further atoms capping the square faces at somewhat larger and even more variable M-N distances. Figure 5 b shows a chain-like connection of trigonal prism by sharing edges. Dubois 90 proposed the packing rule of TTP based on the theory of chemical twinning. Gaskell 87 used this model to calculate the RDF of Pd-Si and found that it agreed well with the experimental data.

However, there exist some limitations for describing the structure of metallic glasses 91 - Boudreaux and Frost 92 found two different structures octahedron and trigonal prism in Pd-Si, Fe-P and Fe-B with the help of computer simulation. So the TTP unit is not the only structure in metallic glasses. In addition, the model assumed that the neighbor bond length and angle are unchanged which make the system a smaller density than the corresponding component crystal.

Thus it does not meet the principle of dense packing of metallic glasses, and the difference of densities obtained by theoretical calculation and experiment is obvious 94 , The model not only shows the homogeneity of metallic glasses macroscopicly, but also explains why the structure of the metallic glasses has long-range disorder. It reveals many properties such as isotropy, but cannot explain micro-inhomogeneity and phase splitting phenomenon of metallic glasses. In the s, Wang 96 suggested that amorphous alloys may have a crystal-like, short-range structure which retained by stacking various types of polyhedra and atomic disorders at random.

These polyhedrons are called clusters those are considered as the basic unit in the structure of metallic glasses. They have more abundant atomic connection compared with the five basic Bernal polyhedrons 97 , In addition, the five-fold symmetry of the clusters can prevent the way of crystal growth to a certain extent. This is consistent with the principles of short-range order and long-range disorder in metallic glasses. He put many hard spheres in a three-dimensional box and then made the spheres dense packing by shaking.

He found that there exist many stable icosahedron clusters stacking in fcc. In this model, the center atom of cluster is regarded as solute atoms, and these atoms which occupied the gap of cluster are regarded as solvent atom. The model considers only three topologically distinct solutes and these solutes have specific and predictable sizes relative to the solvent atoms , Figure 6 a shows an plane of clusters in dense cluster-packing structure, which illustrates the feature of interpenetrating clusters and efficient atomic packing around each solute.

In this model, the solute-centered clusters with the solvent efficiently packed were considered as the basic building blocks and represented the SRO in metallic glasses. Furthermore, these clusters connected with each other by sharing the solvent atoms and formed the MRO. Figure 6 b shows the 3D Miracle dense cluster model for metallic glasses. Figure 6. In Miracle's model, the order of the cluster-forming solutes cannot extend beyond a few cluster diameters, and hence the characteristic of disorder can be retained beyond the nanoscale.

However, the model includes defects that provide rich structural description of metallic glasses. It could not only predict the number of solute atoms in the first coordination shell of a typical solvent atom, but also provide a remarkable ability to predict metallic-glass composition accurately for a wide range of simple and complex alloys. In general, this model is more successful in recent studies on glassy alloys. According to the model of Miracle, Sheng et al.

They confirmed the dominance of solute-centered clusters and put forward a dense equivalent cluster packing model. They thought that the basic unit of metallic glasses is cluster made up of a variety of Voronoi polyhedra. Meanwhile they used Voronoi index method to measure the different clusters, then process statistical analysis.

The polyhedron structure in several representative metallic glasses is listed in Figure 7 b. The packing of solute-centered icosahedron cluster obtained by using the common-neighbor analysis is displayed in Figure 8. It can be seen from this figure that the majorities of clusters are icosahedron and icosahedron-like and clusters in Ni 81 P 19 , Ni 80 P 20 and Zr 84 Pt Figure 8 b c d show the three typical cluster connections.

In the same system, the type of cluster is similar, especially the type of topology and the coordination number. Figure 7. Figure 8. The dense equivalent cluster packing model validates the important role of the effective atomic size ratio between the solute and solvent atoms and explicitly specifies the packing topologies for various CNs.

It greatly reduces the human subjective factor and the system error and enhances people's understanding of structure of metallic glasses. Recently, based on the study of the Cu-Zr metallic glasses, Almyras et al. They demonstrated that the combinations of icosahedral-like clusters can follow a sequence of magic number by considering the system's stoichiometry. In experiments, the combinations of icosahedral-like clusters were referred to as "superclusters" and they regarded superclusters as basic building units Figure 9 shows a case of atom supercluster composed of two atom icosahedrons-like clusters, in which the brown spheres represent copper and the blue represent zirconium atoms.

Their experimental results well agreed with the simulation and could reproduce the structural characteristics of the system. They suggested that the interconnections of clusters and consequently the formation of the superclusters were topological This method may be used to interpret the design of new metallic glasses and to understand their experimental data