Close Packing in Three Dimensions

Dhristi JEE 2022-24

Three-Dimensional Close Packing Meaning

Three-Dimensional close packing in solids is referred to as putting the second square closed packing exactly above the first. It is possible to form Three-Dimensional close packing. In this tight packing, the spheres are horizontally and vertically correctly balanced. Similarly, we can obtain a simple Cubic lattice by adding more layers, one above the other. This can be done in two ways:

  • Three-Dimensional close packing from two-Dimensional square close-packed layers: By putting the second square closed packing exactly above the first, it is possible to form Three-Dimensional close packing. In this tight packing, the spheres are horizontally and vertically correctly balanced. Similarly, we can obtain a simple Cubic lattice by adding more layers, one above the other.

  • Three-Dimensional close packing from two-Dimensional Hexagonal close-packed layers: With the assistance of two-Dimensional Hexagonal packed layers, Three-Dimensional close packing can be shaped in two ways:

  1. Stacking over the first layer on the second layer

  2. Stacking over the second layer on the third layer

This is the space lattice's actual structure. It occurs because of the unit cells' Three-Dimensional structure. Now, the continuous and repeated stacking of the two-Dimensional structures above each other shapes this structure. It can happen in two ways:

Hexagonal Closest Packing: Here, the alternating layers fill the distance between each other. In one layer, spheres align to fit into the holes of the previous layer. There is the same alignment for the first and the third line. So we call this sort of ABA type.

Cubic Closest Packing: The layers are arranged in symmetry exactly above each other here. This form takes the shape of a cube, hence the name. The coordination number of a system of this kind is 12.

The word "closest packed structures" refers to the crystal structures (lattices) with the most tightly packed or space-efficient composition. The spheres must be arranged as close as possible to each other to maximize the efficiency of packing and minimize the amount of unfilled space. Let us retain the Hexagonal close packing in the first layer to establish Three-Dimensional close packings. Each sphere in the second layer rests in the hollow at the center of the Three touching spheres in the layer for near packing, as shown in the figure. 

Strong lines represent the spheres in the first layer, while split lines show those in the second layer. It should be remembered that the spheres in the second layer (either b or c) occupy just half of the triangular voids in the first layer. In the first sheet, unoccupied hollows or voids are indicated by (c). It is found that a tetrahedral void is formed wherever a sphere of the second layer is above the void of the first layer. Whereas in other cases, it is found that the triangular voids in the second layer are above the triangular voids in the first layer, in such a way that the triangular shapes of these voids do not overlap. These voids are classified as octahedral voids and are surrounded by six spheres.

We can calculate the number of these types of voids in the following way:

Let the number of close-packed spheres be N, then

The number of octahedral voids generated = N

The number of tetrahedral voids generated = 2N

From this, we can conclude that the number of octahedral voids generated is equal to the number of close-packed spheres. The number of tetrahedral voids generated is equal to 2 times the number of close-packed spheres.

Close Packing in Crystals

Close packing in crystals is also referred to as close packing in solids which is defined as the efficient arrangement of constituent particles in a crystal lattice in a vacuum. We have to assume that all particles (atoms, molecules, and ions) are of the same spherical solid form to understand this set more clearly. So a Cubic shape is the unit cell of a lattice. Now, there will still be some empty spaces when we stack spheres in the cell. The arrangement of these spheres has to be very effective to reduce these empty spaces. To minimize empty spaces, the spheres should be arranged as close together as possible. 

The definition of Coordination Number is another connected one. In a crystal lattice structure, the coordination number is the number of atoms that surround a central atom. It is often referred to as LIGANCE. Inside a unit cell, the most powerful conformation atomic spheres can take is known as the nearest packing configuration. In two modes, there are densely packed atomic spheres: Hexagonal closest packing (HCP) and Cubic closest packing (CCP).

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FAQs (Frequently Asked Questions)

1. What is packing efficiency and how do you calculate it?

The packing efficiency is the fraction of the atoms currently occupied by the crystal (or unit cell). It must always be less than 100 percent because spheres can not be filled without any space between them (atoms are normally spherical).

The packing efficiency can be calculated using the following steps:

  1. First, we need to calculate the volume of the unit cell.

  2. Count how many atoms per unit cell are there.

  3. Calculate a single atom's volume and multiply it by the number of atoms in the cell of the unit.

  4. Divide this outcome by the unit cell's volume.

2. What is the coordination number in three-dimensional close packing?

The spheres of the third layer are not associated with those of either the first or the second layer in this sort of tight packing. Therefore, if the first layer is called 'A' and the second layer is named 'B,' the third layer is named 'C.' You may write this pattern of layers as an ABCABC. This crystal structure is called the structure of cubic close-packed (ccp) or face-centered cubic (fcc). In each of them, since each sphere is in contact with twelve spheres, the coordination number is 12. In these kinds of close packing, 74% of the space in the crystal is packed.

3. What is 3D packing ?

A Three-Dimensional intertwined circuit (3D IC) is a MOS ( essence-oxide semiconductor) integrated circuit (IC) manufactured by mounding silicon wafers or dies along with hitching them vertically using, for case, through-silicon vias (TSVs) or Cu-Cu connections, so that they bear as a single device to achieve performance advancements at reduced power and lower footmark than conventional two Dimensional processes. The 3D IC is one among various 3D integration schemes that exploit the z- direction to realize electrical performance helps in microelectronics and nanoelectronics.

 3D integrated circuits are often classified by their position of interconnect scale at the worldwide ( package), intermediate ( bond pad) and original (transistor) position. (1) In general, 3D integration is a broad term that includes similar technologies as 3D wafer- position packaging (3DWLP);2.5 D and 3D interceder- grounded integration; 3D piled ICs (3D-SICs); monolithic 3D ICs; 3D miscellaneous integration; and 3D systems integration

4. Describe Complementary metal-oxide-semiconductor (CMOS).

Reciprocal metal – oxide – semiconductor (CMOS, pronounced" see-moss"), also known as reciprocal- harmony essence – oxide – semiconductor (COS-MOS), is a type of essence – oxide – semiconductor field- effect transistor (MOSFET) fabrication process that uses reciprocal and symmetrical dyads of p- type and n- type MOSFETs for sense functions. (1) CMOS technology is employed for constructing intertwined circuit (IC) chips, as well as microprocessors, microcontrollers, memory chips ( including CMOS BIOS), and other digital sense circuits. CMOS technology is also used for analog circuits similar as image detectors (CMOS detectors), data transformers, RF circuits (RF CMOS), and largely intertwined transceivers for numerous types of communication.

5. What is hexagonal close packing?

In Hexagonal close quilting, layers of spheres are packed so that spheres in interspersing layers overlie one another. As in boxy close quilting, each sphere is girdled by 12 other spheres. Taking a collection of 13 similar spheres gives the cluster illustrated over. Bringing together the centers of the external 12 spheres gives Johnson solid J, (27) known as the triangular orthobicupola (Steinhaus 1999,pp. 203-205; Wells 1991,p. 237). 

Hexagonal close quilting must give the same quilting viscosity as boxy close quilting, since sliding one distance of spheres can not affect the volume they enthrall. To corroborate this, construct a Three-Dimensional illustration containing a Hexagonal unit cell with Three layers (Steinhaus 1999,pp. 203-204). Both the top and the bottom contain six 1/ 6- spheres and one semicircle. 

6. What is random close packing?

The conception of " arbitrary close quilting" was shown by Torquato etal. (2000) to be a Mathematically ill- defined idea that's better replaced by the notion of"maximally arbitrary jammed."

Random close quilting of spheres in Three confines gives a quilting viscosity of only eta approx0.64 (Jaeger and Nagel 1992), significantly lower than the optimal quilting viscosity for boxy or Hexagonal close quilting of0.74048. 

Donev etal. (2004) showed that a maximally arbitrary jammed state of M&M's chocolate delicacies has a quilting viscosity of about 68, or 4 lesser than spheres. Likewise, Donev etal. (2004) also showed by computer simulations other ellipsoid paddings redounded in arbitrary quilting consistence approaching that of the thick sphere paddings, i.e., filling nearly 74% of space. 

7. What is sphere packing?

Define the quilting viscosity eta of quilting of spheres to be the bit of a volume filled by the spheres. In Three confines, there are Three periodic paddings for identical spheres: boxy chassis, face-centered body chassis, and Hexagonal chassis. It was hypothesized by Kepler in 1611 that close quilting ( boxy or Hexagonal, which have original quilting consistency) is the thick possible, and this assertion is known as the Kepler guess. 

In 1831, Gauss managed to prove that the face-centered boxy is the thick chassis quilting in Three confines (Conway and Sloane 1993,p. 9), but the general guess remained open for numerous decades. 

While the Kepler guess is intimately egregious, the evidence remained unexpectedly fugitive. Rogers (1958), a well-known experimenter on the problem, remarked that" numerous Mathematicians believe, and all Physicists know"that the factual answer is 74.048 (Conway and Sloane 1993,p. 3)