Close Packing in Crystalline Solids in Two Dimension

A diagrammatic depiction of the three-dimensional configurations of component particles in a crystal, known as the crystal lattice, in which each constituent particle is shown as a point in space. There is extremely little room between the atoms that make up a crystal lattice because the atoms are packed so closely together in the lattice. As a direct consequence of this, the unit cell of a lattice takes on a cubic shape. When we are finished stacking the spheres, the cell will still have some open space within it. In order to get rid of these empty areas, the organisation of these spheres needs to be exceedingly efficient. It is important to minimise the amount of space between the spheres, so position them as closely as you can. 

Coordination Number

There is also a connection to the concept of the Coordination Number. The number of atoms that surround an atom in the centre of a crystal lattice is referred to as the coordination number for that structure. As a direct result of this, there are three different directions in which the component particles are packed together very closely. In the following sections, tight packing in crystals and solids in various dimensions is discussed: Packing Density Close to Capacity in One Dimension When packing in one dimension, spheres are arranged in a row in such a way that neighbouring atoms are brought into touch with one another. The number of particles that are immediately next to one another is referred to as the coordination number. In the scenario of one-dimensional tight packing, the coordination number is equal to two. 

Tight Packing in a Two-Dimensional Space 

In two-dimensional close packing, a pattern in two dimensions is created by stacking rows of closed-packed spheres one on top of the other. There are two different ways to stack items: Packing in a Square Closed Square: When squeezing things in tightly, you can position the second row just below the first row. Because of this, if we refer to the first row as a “A” type row, then the second row, which is structured in the exact same manner as the first row, is likewise a “A” type row. In this particular arrangement, each sphere is in touch with a total of four other spheres. As a direct consequence of this, its coordination number is four. It has come to our attention that a square is produced when the centres of the four spheres that are geographically closest to one another are brought together. This particular sort of packing is referred to as square close packing in two dimensions when it occurs in crystalline materials.

Close-Packing in a hexagonal pattern (an ABAB type arrangement)

 It is possible to stagger the placement of the second row below the first row, with the spheres of the second row fitting into the depressions of the first row. If we refer to the first row as an “A” type row, then the second row, which is structured in a different way, may be referred to as a “B” type row.Another instance of the “A” form appears in the third row. The “ABAB” packing style is the name given to this particular arrangement. In this particular arrangement, each sphere is in touch with a total of six other spheres. A hexagon is created when the centres of six spheres that are immediately next to one another are joined to one another. This kind of solid packing is known as hexagonal close packing in two dimensions, and that is the name given to it. It differs from square close packing in that it has a smaller amount of open space and, as a result, a greater packing efficiency.

The Three-Dimensional Close-Packing Method 

Packing materials closely together in three dimensions results in the formation of true lattices and structures. They are made by constructing successive layers of two-dimensional spheres and stacking them atop one another. There are two routes that may be used to reach this goal: The two-dimensional square close-packed layers serve as the basis for the three-dimensional tight packing. The three-dimensional extension of the tight packing found in the two-dimensional hexagonal layers From Two-Dimensional Close-Packing in Crystals to Three-Dimensional Close-Packing: The formation of three-dimensional close packing in solids can be accomplished by positioning the second square closed packing directly above the first. In this dense packing, the spheres are arranged in the correct orientation both horizontally and vertically. In a similar manner, we may construct a straightforward cubic lattice by piling more layers on top of one another. The unit cell that makes up the basic cubic lattice is referred to as the primitive cubic unit cell. 

From Two-Dimensional Hexagonal Close-Packing in Crystals to Three-Dimensional Close-Packing in Crystals

The three-dimensional extension of the tight packing found in the two-dimensional hexagonal layers The support of two-dimensional layers packed in a hexagonal pattern allows for the formation of three-dimensional tight packing in two different ways: Putting the second layer on top of the first layer in a stacked configuration Placing the third layer on top of the second layer in a stacked configuration. Putting Together the Second Layer On Top of the Initial Layer Assume we take two hexagonally tight-packed layers ‘A’ and position them over the second layer B (because the spheres in both layers are aligned differently) in such a way that the spheres of the second layer are positioned in the depressions of the first layer. This would indicate that the spheres in the second layer are aligned differently than the spheres in the first layer. A tetrahedral void is produced whenever a sphere from the second layer is positioned immediately above the void (space) produced by the first layer. We also see octahedral voids at the spots where the triangular voids of the second layer are positioned close to the triangular voids of the first layer in such a way that the space occupied by the triangles does not overlap. The octahedral cavities are surrounded by six spheres. 

Conclusion

In three-dimensional packing, the spheres that make up the third layer are arranged so that they are directly above the spheres that make up the first layer. If we were to assign the letter A to the first layer and the letter B to the second layer, the pattern would look like this: ABAB… and so on. The structure that was produced as a result is referred to as an HCP, which stands for hexagonal close-packed structure. When it comes to this particular style of packing, spheres are not included in either the second layer or the first layer. If we refer to the first layer as A, the second layer as B, and the third layer as C, the pattern would be represented by the letters ABCABC (as it is now a separate layer). The structure that is formed as a consequence is sometimes referred to as a cubic close packed (ccp) structure or a face-centered packed cubic structure (fcc). Crystallization occurs in the structure of certain metals, such as copper and iron, for example. Due to the fact that every sphere in the system is in direct communication with 12 other spheres, the coordination number for both scenarios is 12. It should be noted that the packing is incredibly efficient, with around 74% of the crystal being utilised in its entirety.